Emergency services rescued about 2,000 people stranded after Hurricane Harvey battered the Texas coastline, killing two people and injuring at least 14.
Those numbers are expected to rise as local authorities on Sunday ordered people to stay indoors.
It is the worst storm to hit the area in 50 years.
There are reports of possible deaths in submerged vehicles, but investigations continue, Chief Darryl Coleman of the Harris County Sheriff's Office said.
The fear now is from catastrophic flooding as Harvey dumps torrential rain over south-east Texas.
President Donald Trump will visit Houston on Tuesday, Reuters reports.
The National Weather Service (NWS) said 12 tornadoes had been reported across the region and that conditions are "unprecedented".
It said there was a "flash flood emergency" across the Houston Metro area, with travel near impossible.
One of the two confirmed fatalities was a woman in Houston who drove her vehicle into high water, according to city police who said they believe the water was too high to pass and her car became inoperable.
When the victim got out of her vehicle, she was swept away by floodwater and drowned. Her body was found by neighbours.
“There’s flooding all over this city,” said Art Acevedo, Houston police chief, as he appealed for people to stay at home. “We have one fatality, and a potential second fatality from the flood waters out here.”
An earlier death was reported on Friday night when a man died in a house fire in the town of Rockport close to where Harvey made landfall.
The fiercest hurricane to hit the US in more than a decade came ashore late on Friday night about 48 kilometres north-east of the town of Corpus Christi. It arrived with winds gusting at 130mph and was classified as a category four storm – the second highest level.
It shredded trees and destroyed roofs as it made its way slowly inland, reducing in force as it moved away from the energy-giving warm waters of the Gulf of Mexico.
A dozen recreational vehicles were flipped on their side in Rockport where a convoy of military vehicles arrived on Saturday to help with recovery efforts.
“It was terrible,” Joel Valdez, 57, told Reuters. The storm ripped part of the roof from his trailer home at about 4am, he said as he huddled in his Jeep for safety. "I could feel the whole house move."
The winds had eased by Sunday morning. But as the first assault ended, its second began.
With the storm almost stationary it dumped torrential rain over a wide area that included Houston, America’s fourth most populous city, and a flood-prone place at the best of times.
“There are a number of stranded people on our streets, calling 911, exhausting needed resources. You can help by staying off the streets,” Sylvester Turner, the mayor of Houston, said on Twitter.
Mr Turner has urged residents not to call emergency services unless their situation is life-threatening and they need to be rescued.
As the hurricane was downgraded to a tropical storm, he warned: "Don't get on the road. Don't assume this storm is over.W
Rain fell on the city at a rate of almost three inches an hour, leaving streets and underpasses submerged in water. As much as 40 inches is forecast to fall.
In some areas, residents were advised to seek refuge on their roofs.
Jeff Lindner, flood officer with Harris County, which includes Houston, said people inundated by rising water should not crawl into attics but get on top of them.
Greg Postel, meteorologist and hurricane specialist for The Weather Channel, said with waters rising fast in the Houston area it "could be the worst flooding disaster in US history".
How will Hurricane Harvey affect US energy sector?
Hurricane Harvey: Experts warn of ‘life-threatening’ flooding
The storm cut power to more than 300,000 people and triggered hundreds of rescues. In Houston, police went door-to-door in two flooded apartment complexes helping residents to safety.
Oil and gas production were largely halted, sending fuel prices soaring.
About 22 percent of U.S. Gulf of Mexico oil production is offline due to Tropical Storm Harvey, the U.S. Department of the Interior's Bureau of Safety and Environmental Enforcement (BSEE) said on Sunday.
Output levels rose slightly from Saturday, when roughly 25 percent of Gulf oil output was offline.
The amount of production offline on Sunday was roughly 378,633 barrels of oil per day out of the roughly 1.75 million bpd pumped from the Gulf.
The sense of crisis was clearly visible on US officials' Twitter feeds as they dealt with multiple calls for help.
Ed Gonzales, Harris County Sheriff, made multiple requests for people to “shelter in place” while calling emergency services to a woman going into a labour, someone suffering a heart attack, reports of a woman and child trapped in a submerged car, and another woman who posted: “I have 2 children with me and water is swallowing us up.”
“Getting lots of requests for hi-water rescues,” he wrote at one point. “Some involve children, others with medical issues. Trying to get to as many as possible.”
The hurricane poses the first major emergency management test of Trump's administration.
He has been monitoring developments from Camp David where he signed a disaster proclamation, freeing federal aid to help emergency efforts.
He also met cabinet and staff on Saturday to discuss the federal reaction to the storm, according to the White House.
“Great co-ordination between agencies at all levels of government. Continuing rains and flash floods are being dealt with. Thousands rescued,” he wrote on Twitter on Sunday morning.
“Many people are now saying that this is the worst storm/hurricane they have ever seen. Good news is that we have great talent on the ground.”
He initially said he planned to visit as soon as he could do so without distracting from the emergency operation.
Meanwhile, the US Coast Guard said it had rescued 20 people from distressed vessels on Saturday, and was monitoring two cruise liners carrying thousands of people stranded in the Gulf of Mexico.