[Updated at 8:04 p.m. ET] Triumph passenger Sylvester Davis, backing up what other passengers have said today, told CNN by phone that the ship was cleaner today.
“It got a lot better once the notification went out that the (National Transportation Safety Board) and the Coast Guard was coming on board," Davis said. "They spent a lot of time cleaning up, and it's sort of frustrating because it doesn't look like now the way it’s been looking and the way we've been living.”
[Updated at 7:36 p.m. ET] We've received video, taken Sunday morning, showing smoke from the fire that knocked out the ship's power. The video was sent to us by passenger Megan Clemons-Foxall.
[Updated at 7:26 p.m. ET] The end of the trip is near: The ship is about 17 miles from the dock. We're anticipating an arrival between 10 p.m. ET and midnight ET.
It won't be over immediately for the crew and passengers. Because more than 4,000 people are aboard, and there's just one working elevator, Carnival anticipates it'll take about four or five hours to get everyone off the ship.
[Updated at 6:20 p.m. ET] Like other passengers who talked to CNN today, passenger Julie from Lake Charles, Louisiana, tells CNN by phone that the crew members have been great.
"The crew has worked nonstop," she said. "They have been (working) from daylight till dark. Once they got a break, it was too hot in their deck to sleep. Too hot, no air flow.”
Julie, her daughter, her sister and a couple of nieces took the trip together. Their cabin was too hot to sleep in since Sunday's power outage, so they've been sleeping in the hallway outside.
"Miserable. Probably the worst time in our life," Julie said. "We’ve had to put a lot of faith in God."
Food has been available, though some people - worried that the food supply would run out - hoarded, sometimes making it difficult for others to get food immediately, she said.
[Updated at 5:50 p.m. ET] We've been getting a bunch of pictures from Triumph passengers ahead of the ship's docking. Here's one that passenger Megan Clemons-Foxall sent of what she said was "buckets of sewage" in a stairwell aboard the ship.
We wrote earlier about passengers improvising cell-phone charging stations with surge protectors and the few still-working outlets or computers. Here's a look at one of these stations, from passenger Kaitlyn Robertson.
For more pictures, check out this gallery.
[Updated at 5:04 p.m. ET] The Coast Guard has taken one guest off the ship because of a medical issue, out of an abundance of caution, Carnival Senior Vice President Terry Thornton said.
[Updated at 4:36 p.m. ET] CNN has had a helicopter flying over the ship, and passengers are clearly aware. Here's some people arranging themselves on the deck, spelling what appears to be "help."
[Updated at 4:28 p.m. ET] Even after the ship docks tonight or early Friday morning, passengers still will have a while to wait before they can leave.
It could take four to five hours for Carnival to complete the disembarking process, because of the ship's power problems, said Terry Thornton, Carnival senior vice president. The ship has only one working elevator right now, he said.
Thornton says the ship could dock between 10:30 p.m. ET Thursday and 1:30 a.m. ET. Friday.
[Updated at 4:20 p.m. ET] We have a new estimated time of the Triumph's arrival at port in Mobile: From 10:30 p.m. to midnight ET. This comes after a delay precipitated by a busted towing line.
The line was replaced not long ago, and tugboats are again moving the ship.
[Updated at 4:01 p.m. ET] More descriptions of sewage running down the ship's interior walls this week: Triumph passenger Darryl Malone tells CNN that that "it sounded like it was raining" sewage at one point.
Malone echoed what passenger Donna Gutzman told us earlier - things are cleaner today.
"Today they tried to clean it up and make it look better, because everyone (including the Coast Guard) came on board," Malone told CNN's Brooke Baldwin.
[Updated at 3:49 p.m. ET] The cruise ship is moving again, thanks to a replacement tug line, Carnival spokeswoman Joyce Olivia says.
A line between a tugboat and the cruise ship broke this afternoon, so the tugging was halted while a replacement was attached.
Meanwhile, it appears officials are committed to having the Triumph docked tonight.
According to Justina Strong with the Alabama State Shipping Authority, the Triumph is in the shipping channel - that indicates officials do intend to have the ship at port tonight.
[Updated at 3:41 p.m. ET] Triumph passenger Donna Gutzman says that the ship is much cleaner today than it had been.
"This ship was a wreck before (the Coast Guard boarded the ship today)," Gutzman said. "We were walking through urine."
When she woke up this morning, "all the chairs on the deck were lined back up ... (and) the water that was standing has been vacuumed," she told CNN's Brooke Baldwin by phone from the ship.
[Updated at 2:57 p.m. ET] A few more details about our interview with passenger Julie Morgan earlier this hour - she offered some more insight on why she and others people with rooms on the lower floors had been sleeping on the deck over the past few nights.
Morgan's first-floor room had little ventilation after the outage, and was "very hot and extremely smelly."
Sewage "backed up unto our showers," she said. The ship has been listing since the outage, and "water goes into our rooms," she said.
"We don’t really stay down there. We have our luggage up on our beds to stay dry. We're only going down there to change clothes and take a kind-of shower,” she said.
Though people have had to wait in lines for food, people aren't left in hunger. "We're not hungry at all. We're doing fine on that front," Morgan said. The Coast Guard and a sister cruise ship had dropped off supplies this week.
[Updated at 2:48 p.m. ET] Another passenger tells CNN that the crew members have been great.
"The crew has been wonderful. Couldn't have asked for nicer crew members,” passenger Kimberly Ware told CNN's Brooke Baldwin by phone from aboard the ship.
Ware confirmed what Carnival wrote minutes ago on Twitter: The tugging process has been delayed. Carnival wrote that a tug line broke and is being replaced.
The worst part of the trip for Ware was Sunday's fire, she said. Just the day before during muster drills, crew members told passengers that fire was the greatest danger for cruise ships, she said.
"The lowest point for me was waking up and seeing the smoke and glow from the fire. That was very, very frightening to me, and the hours after that until we knew we were safe," Ware said.
[Updated at 2:38 p.m. ET] There's been a hiccup in the towing operation:
[Updated at 2:31 p.m. ET] If you're wondering why some passengers are just now talking to their onshore relatives by phone - in some cases for the first time in days - passenger Julie Morgan offers an explanation:
First, there's the reception situation. Since Sunday's power outage, people generally haven't had cell phone reception except in brief periods of time when other cruise ships came nearby, Morgan told CNN by phone minutes ago. Since the ship is coming close to Alabama's shore, reception is better.
Second, people have struggled to keep their cell phones charged. A generator is on the ship. Passengers have "rigged charging stations from the ship computers," Morgan said. She didn't go into too much detail, but apparently it involves lots of surge protectors.
[Updated at 2:18 p.m. ET] Though the conditions are bad, Triumph passenger Julie Morgan has nothing but compliments for the ship's crew.
"The crew on the ship has been phenomenal," Morgan told CNN by cell phone moments ago. "Never a bad word (from them). They've been working to the bone,” only getting three hours of sleep at a time, she said.
"All our complaints would be with the head office in Miami," Morgan said.
Morgan said she's frustrated about communication with Carnival officials on land - it took her a while to figure out how she was getting home, because the ship had departed from and initially was supposed to return to Houston.
Her husband is meeting her in Mobile. He told CNN that although Carnival is offering to transport passengers to a hotel in New Orleans, he said he plans to stay with his wife for a night in Mobile.
[Updated at 2:13 p.m. ET] It's been an interesting climate change for the passengers and crew - it's 51 degrees in the Mobile, Alabama, area, and it was cooler this morning. Passengers have told us by phone that they've been wearing bathrobes to stay warm today - remember, most didn't pack for a cool day, since it was a cruise to the Caribbean.
But most of the past five days, people on board have been dealing with stifling heat. Last night, passenger Julie Morgan told CNN by cell phone, people were sleeping under sheets on the deck.
[Updated at 1:48 p.m. ET] CNN's Chad Myers reports that strong winds and unfavorable currents still are hampering efforts to tow the stricken cruise ship toward shore. The Carnival Triumph, with the help of four tugboats, is currently moving at just 1 mph.
[Updated at 1:19 p.m. ET] About 30 minutes ago we wrote about getting a father and daughter on the ship to wave to our helicopter camera. The video also showed the pair's friends, including 10-year-old Allie Taylor and her father.
Allie's mother, Kim, who is not on the ship, saw it on CNN. The mother and the daughter talked by phone moments ago, and CNN carried the conversation.
"It’ll be soon baby, very soon. You’ll be here tonight," Kim told her daughter.
Allie told CNN about the sanitary situation on the ship.
"We have to use the restroom in a bag because the water doesn’t work,” she said.
Earlier today, passenger Larry Poret, the friend of Allie's father, told CNN by phone that urine and feces streamed in the halls and down walls after toilet facilities failed, soaking the mattress of a friend of his who was sleeping in a hallway.
A couple dozen toilets on the ship are working, Carnival has said.
[Updated at 1:01 p.m. ET] One man who has relatives on the ship, including his 18-year-old daughter, drove from Indianapolis to Mobile to greet them when the ship finally docks.
The man, Rusty Adkins, talked to CNN's David Mattingly moments ago as he waited on shore for the ship. Adkins had just talked privately to his relatives by phone, and he relayed what his daughter had told him about the conditions aboard.
"The ship listed to one side, so grease and sewage and fluids are building up on one side of the ship," Adkins said. "She said it’s kind of nasty. It's not a very good situation at all."
[Updated at 12:46 p.m. ET] We seem to have managed a remote mother-daughter reunion of sorts. While we've been in contact with passenger Larry Poret and his daughter Rebekah Poret by phone, we got them to wave from the deck to the helicopter, and then got Rebekah to talk on air to her mother, who didn't take the cruise and is waiting for them on land.
The mother, Mary, told Rebekah that she'll be waiting for her with a warm blanket.
Rebekah had a request for her mom:
"Have some McDonald's there, too. We’re hungry!”
[Updated at 12:39 p.m. ET] The ship is now visible from shore in Alabama, but the ship has quite a way to go - it's about 45 miles away, and it still has to navigate a channel, CNN's Ashleigh Banfield reports.
[Updated at 12:33 p.m. ET] More details about the sanitary situation: Carnival has said a couple dozen toilets - for the more than 4,000 people aboard - are working. Triumph passenger Larry Poret tells CNN by phone that the toilets' functionality has been erratic.
"The toilet will flush one time, and the next time you need them to flush they won't. They may not flush for hours and hours," Poret said.
[Updated at 12:29 p.m. ET] Passengers have reported sewage sloshing around in hallways, flooded rooms and trouble getting enough to eat.
Talking to CNN by phone, Triumph passenger Larry Poret says he's not particularly worried about food at this point.
"To be honest with you, we're just tired. ... Just get us out of here."
[Updated at 12:24 p.m. ET] CNN's video shows passengers on the Triumph with makeshift signs saying "The ships afloat" and "Get us to Louisiana."
[Posted at 12:16 p.m. ET] CNN has a helicopter giving us our first live views of the crippled ship Carnival Triumph - where thousands of passengers have been living in squalid conditions - limping toward port in Mobile, Alabama, Thursday afternoon.
It's been a nightmare five days for the 3,143 passengers and 1,086 crew members on board the 900-foot, 14-story Triumph, where a fire broke out in the engine room early Sunday. Much of the ship's electrical power went down in the fire, causing widespread malfunctions, including taking out sanitary systems.
Passengers have been gathering on the deck for days to get away from terrible smells and stifling heat on the inside, though as the ship gets closer to shore, the air is colder, and some are huddling together with whatever blankets or bathrobes they can find.
A passenger, Larry Poret, is on the phone with CNN - he says he's frustrated and just wants off the ship.
“We're frustrated. ... Why can’t we have more tugboats?" he said.