The center of the storm hit St. Vincent Island, Florida, early Saturday and continued to moved in a northeast direction toward toward the Carolinas, with winds up to 50 mph expected along the Atlantic coasts of Florida and Georgia.
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The National Weather Service blamed Nestor for a series of tornadoes that struck Hillsborough and Polk counties in Florida Friday night, damaging an estimated 50 homes.
The tornadoes caused power outages for thousands of customers across central Florida.
"Hurricanes have seen damage, but never have I seen a tornado do something as bad as this," a resident told NBC News affiliate Telemundo 49 in Tampa.
Officials warned residents of 6- to 8-feet surf and rip currents along Florida's Gulf Coast.
"As #Nestor moves north, storms associated with this system are still producing dangerous beach conditions across the Gulf Coast, and in areas of the Atlantic Coast," the Florida Division of Emergency Management tweeted. "Pay attention to beach warning flags and do not swim in dangerous conditions."
The storm could get a slight second wind Sunday when some strengthening is expected, the weather service said.
When it was still a tropical storm Friday night, Nestor bore down on the northern Gulf Coast with high winds, surging seas and heavy rains. At one point, it had threatened to hit an area of the Panhandle devastated one year ago by Hurricane Michael.
Dennis Romero writes for NBC News and is based in Los Angeles.