MOBILE, Alabama (Reuters) - The cruise ship Carnival Triumph was being towed to port in Alabama on Tuesday as passengers complained of sweltering heat and gut-wrenching odors from backed up toilets after a weekend engine fire left it adrift in the Gulf of Mexico.

The 893-foot (272-meter) vessel has been without propulsion and running on emergency generator power since Sunday when it was disabled off southern Mexico's Yucatan Peninsula. It was carrying 3,143 passengers and 1,086 crew. There have been no reports of injuries.

Brent Nutt of Texas told the CNN network that his wife Bethany, a passenger on the ship, described in a telephone call on Monday "horrific" conditions and fighting over scarce food.

"The odor is so bad that it's making them sick," Nutt said. "They're vomiting and stuff all over the boat just from the odor ... There's feces all over the floor."

The ship, operated by Carnival Corp, left Galveston, Texas, on Thursday and was due to return there on Monday. It was being towed by tugboats to Mobile, Alabama, and expected by the company to arrive there no later than Thursday.

Most of the 23 public restrooms on the ship are working along with some toilets in guest cabins. Passengers also have access to running water for showers, said Carnival Cruise Lines President and Chief Executive Gerry Cahill.

"There is no question that conditions aboard the ship are very challenging," Cahill told reporters in Miami.

"Let me assure you, no one from Carnival is happy about the conditions on the ship and we obviously are very, very sorry," he said.

Passenger Ann Barlow told CNN the smell from backed-up sewage was "overwhelming." "It's just disgusting," she said. "It's the worst thing ever."

"There's no air conditioning at all inside," said Barlow, adding that some passengers were sleeping outdoors and staying on an upper-level deck for fresh air.

Barlow said crew members were distributing free beer and wine. "Everybody's just trying to do the best they can," she said.

Passengers would receive a full refund for the cruise plus transportation expenses and a future cruise credit equal to the amount paid for this voyage, said Cahill in a statement.

They would also be reimbursed for all shipboard purchases except for gift shop and casino charges.

The company has reserved more than 1,500 hotel rooms in Mobile and New Orleans for passengers when they reach shore and arranged 20 charter flights along with buses and other transportation, Cahill told reporters.

The problems on the ship have given Miami-based Carnival Corp another public relations nightmare.

In January 2012, the cruise ship Costa Concordia, operated by Carnival-owned Costa Cruises, struck rocks and was grounded on its side off the Tuscan island of Giglio in Italy. Thirty-two people were killed in that disaster.

In November 2010, an engine fire on the Carnival Splendor crippled its propulsion system and knocked out most of its power while it was off the Pacific coast of Mexico.