Florence was downgraded to a tropical depression on Sunday as it continued its slog west over the Carolinas, flooding roads and homes, downing trees and power lines and sending thousands to shelters. The storm has been blamed for at least a dozen deaths.

But even as the storm itself weakened, the National Hurricane Center warned early Sunday that "flash flooding and major river flooding will continue over a significant portion of the Carolinas."

More than 400 people were were rescued from floodwaters in New Bern, North Carolina, and officials warned that the flooding risk was not over. Rivers in North Carolina could crest as late as Wednesday, said Gov. Roy Cooper.

"More people now face imminent threat than when the storm was just offshore," Cooper said. "I cannot overstate it: Floodwaters are rising, and if you aren’t watching for them, you are risking your life — even in places that don't typically flood."

New Bern, North Carolina, Mayor Dana Outlaw told "Meet the Press" on Sunday that officials remained "very concerned" about additional flooding and trees still toppling.

"A lot of the creeks around New Bern are increasing by the hour," he said.

Outlaw urged residents to stay inside and follow the local curfew as 30 roads were still impassible. He added that 4,200 homes had been damaged, some extensively, and 300 commercial buildings had also been damaged.

The National Hurricane Center warned that parts of central and western North Carolina and far southwest Virginia could receive an additional 5 to 10 inches of rain. There could be a total storm rainfall of 20 inches in western North Carolina and up to 40 inches in the southeastern part of the state.

"These rainfall amounts will produce catastrophic flash flooding, prolonged significant river flooding, and an elevated risk for landslides in western North Carolina and far southwest Virginia," the hurricane center said in a 5 a.m. bulletin.

The center added that tornadoes remained possible across North Carolina and eastern South Carolina on Sunday. Swells generated by Florence are also affecting portions of the east coast and Bermuda and could cause life-threatening surf and rip current conditions.

The U.S. Coast Guard said that 20 to 30 people were rescued Saturday from a subdivision near Jacksonville, North Carolina. Kevin Futrell, a 43-year-old glass cutter who used a boat to help people escape the flooded River Hills subdivision, said he could see "the fear in their eyes." His boat struggled in the strong currents.

"My boat stalled twice and I could definitely see the people in my boat were worried," Futrell said. "I knew I had someone else's life in my hands." He got around 15 people to higher ground near his sister's house, and about three of them were airlifted by the Coast Guard for medical issues.