NEW BERN, N.C.—Florence drenched the Carolinas as it lumbered across the region, claiming at least 14 lives and stranding hundreds of people in floodwaters that were expected to continue to rise.

The storm moved at a 3-mile-an-hour crawl over eastern South Carolina on Saturday and was forecast to soak the Carolinas throughout the weekend, dumping as much as 40 inches of rain along the coast south of Cape Hatteras, N.C. The deluge triggered “catastrophic flash flooding” and significant river flooding, the hurricane center said, and some rivers weren’t expected to crest until early next week.

A preliminary report from Swansboro, N.C., showed that more than 30 inches of rain had fallen so far, according to the National Hurricane Center. That breaks the tropical cyclone rainfall record of 24.06 inches for North Carolina, which was set during Hurricane Floyd in 1999, it added.

“The rainfall is epic, and will continue to be,” said North Carolina Gov. Roy Cooper at a news conference. “We face walls of water, at our coast, along our rivers, across farmland, in our cities and in our towns.”

In North Carolina, the deaths include a mother and infant killed in Wilmington, when a tree fell on a home, authorities said. A woman also died when a fallen tree prevented emergency crews from reaching her while she suffered a heart attack, authorities said.

“ The rainfall is epic, and will continue to be. We face walls of water, at our coast, along our rivers, across farmland, in our cities and in our towns. ” — North Carolina Gov. Roy Cooper

The first storm-related death reported in South Carolina occurred Friday night, when a 61-year-old woman driving on a highway crashed into a toppled tree. Authorities in the state later reported two more deaths from carbon-monoxide poisoning caused by a generator.

More than 730,000 customers lacked power in North Carolina, and another 120,000 outages were reported in South Carolina, according to emergency-management agencies. More than 28,000 people were staying in 220 shelters in the Carolinas.

Mr. Cooper said dangers from flooding were increasing, with some rivers not expected to reach their highest points until early next week. “The water is rising fast everywhere, even in places that don’t typically flood,” he said. “Many people who think that the storm has missed them have yet to see its threat.”

Seven rivers in North Carolina are forecast to reach major flooding levels in the coming days, according to a report by Rick Neuherz, a hydrologist with the National Weather Service. Some, like the Cape Fear River and Lumber River, could set flooding records, he wrote.

On Saturday, Cumberland County, the city of Fayetteville and the town of Wade issued a mandatory evacuation order for all people within a mile of the banks of the Cape Fear and Little Rivers because of flooding risk.

James Trogdon, the state’s transportation secretary, said numerous counties had closed primary roads, and portions of interstate highways were shut. Interstate 95 was closed near Fayetteville, as was Interstate 40 between Wilmington and I-95.

Major Flooding VIRGINIA TENNESSEE Winston-Salem 5 Raleigh Asheville 4 NORTH CAROLINA 1 Charlotte 2 3 Greenville SOUTH CAROLINA Cumulative rainfall, Thurs. through 8:00 a.m. Saturday Wilmington Columbia 5 10 2 inches 20 15 Flood gauge status Major flooding Saturday Major flooding expected through Tuesday GEORGIA Charleston Each dot = 1,000 people Gauges recording major flooding Saturday Flood stage Pollocksville Trenton Chinquapin 3 1 2 20 feet 20 feet 20 feet 15 15 15 10 10 10 5 5 5 0 0 0 Fri. Thurs. Sat. Fri. Sat. Thurs. Sat. Fri. Thurs. Belhaven Streets Ferry 5 4 10 feet 10 feet 5 5 0 0 Fri. Sat. Thurs. Thurs. Sat. Fri. VIRGINIA Winston-Salem 5 Raleigh 4 NORTH CAROLINA 1 Charlotte 2 3 SOUTH CAROLINA Wilmington Cumulative rainfall, Thurs. through 8:00 a.m. Saturday Columbia 2 inches 5 20 15 10 Flood gauge status Major flooding Saturday Major flooding expected through Tuesday Charleston Each dot = 1,000 people Gauges recording major flooding Saturday Flood stage Chinquapin Trenton 1 2 20 feet 20 feet 15 15 10 10 5 5 0 0 Sat. Fri. Thurs. Thurs. Fri. Sat. Streets Ferry Pollocksville 3 4 15 feet 15 feet 10 10 5 5 0 0 Sat. Thurs. Fri. Sat. Fri. Thurs. Belhaven 5 10 feet 5 0 Fri. Thurs. Sat. VIRGINIA Winston-Salem 5 Raleigh 4 NORTH CAROLINA 1 Charlotte 2 3 SOUTH CAROLINA Wilmington Cumulative rainfall, Thurs. through 8:00 a.m. Saturday Columbia 2 inches 20 10 15 5 Flood gauge status Major flooding Saturday Major flooding expected through Tuesday Charleston Each dot = 1,000 people Gauges recording major flooding Saturday Flood stage Chinquapin Trenton 1 2 20 feet 20 feet 15 15 10 10 5 5 0 0 Thurs. Fri. Sat. Sat. Thurs. Fri. Streets Ferry Pollocksville 4 3 15 feet 15 feet 10 10 5 5 0 0 Fri. Thurs. Sat. Sat. Fri. Thurs. Belhaven 5 10 feet 5 0 Thurs. Sat. Fri. Cumulative rainfall, Thursday-8:00 a.m. Sat. 15 10 20 5 2 inches Flood gauge status Major flooding Saturday Major flooding expected through Tuesday Each dot = 1,000 people VIRGINIA NORTH CAROLINA 5 1 Raleigh 4 Charlotte 2 3 Wilmington Columbia SOUTH CAROLINA Charleston Gauges recording major flooding Saturday Passed flood stage Level Saturday 25 feet 1. Trenton 20 2. Chinquapin 15 3. Pollocksville 10 1 4. Streets Ferry 5 5. Belhaven 0 Sat. Fri. Thurs.

“Road conditions across nearly all of our state will be rapidly deteriorating in the coming days,” Mr. Trogdon said. In his entire career, he said, he had “never seen flash flooding like this in our state.”

Federal officials said Saturday that while Florence had weakened, its dangers hadn’t. “What we’re dealing with here is a major flooding and rain event. At this point, we’re also looking at potential landslides,” Neil Jacobs of the National Oceanic and Atmospheric Administration told a news conference.

Jeffrey Byard, associate director at the Federal Emergency Management Agency, said Florence would be costly, in part because of the sheer volume of water it is unleashing and its slow movement.

The NHC downgraded Florence to a tropical storm on Friday and expected to further downgrade it to a tropical depression by Saturday night.

Homeowners and their friends make their way to their vehicle on Saturday after checking on a home that is surrounded by flood waters in Southport, N.C. Photo: Joe Raedle/Getty Images

President Trump approved a major disaster declaration on Saturday, freeing up federal funding for affected areas. He is expected to travel to areas affected by the storm next week, the White House said Friday.

North Carolina’s agriculture industry and environmental groups are closely watching for signs of heavy rains breaching hog waste “lagoons,” which could pose a threat to drinking water. As of late Friday, the North Carolina Pork Council said rainfall hadn’t exceeded the lagoons’ capacity.

Smithfield Foods Inc., the largest U.S. pork producer, said it was assessing the impact on its farms. Smithfield’s slaughtering and processing plants so far appear undamaged, but for the company’s farms, “it’ll take some time before we’ll be able to visit every site,” a spokeswoman said.

In Wilmington, hundreds of trees, including large oaks, collapsed, blocking roadways. Long lines formed at the sole grocery store to open on College Road, near the University of North Carolina at Wilmington.

Sabrina Wood, a 36-year-old mother of two, said she needed milk, bread and beer. She waited in line about 30 minutes in the rain but said it was a welcome respite after being cooped up for 48 hours in a small house with bored children.

Powerless As of Saturday evening, nearly 9 percent of the more than 5 million customers served by major electricity providers in the Carolinas remained without power. Percent reporting power outages 12% 10 8 6 4 7:00 a.m. Friday Hurricane Florence makes landfall 2 Midnight Friday Noon Noon Midnight Saturday

“I’ve been through plenty of storms, but this is the longest one I’ve been through,” Ms. Wood said. “It’s treacherous. It’s stressful.”

So far, South Carolina appeared to be faring better. Gov. Henry McMaster lifted evacuation orders for all affected counties except two: Horry and Georgetown, in the Myrtle Beach area.

But officials warned that residents needed to remain vigilant for possible flooding in the coming days. They were keeping close watch in particular on the Pee Dee River Basin, in the northeastern part of the state.

—Ben Kesling, Jacob Bunge and Rebecca Davis O’Brien contributed to this article.

Write to Arian Campo-Flores at arian.campo-flores@wsj.com, Valerie Bauerlein at valerie.bauerlein@wsj.com and Erin Ailworth at Erin.Ailworth@wsj.com