"A turn toward the east-northeast is expected tonight and Sunday", NHC forecaster Jack Beven said.
Claudet was declared well organized to qualify as a named tropical cyclone early Saturday morning, long after the center of the storm cycle landed southwest of New Orleans.
Tornadoes are possible tonight across southeast Alabama, the western Florida Panhandle, and southwest Georgia.
Sheriff Heath Jackson in Escambia County said a suspected tornado "pretty much leveled" a mobile home park, toppled trees onto houses and ripped the roof off of a high school gym. Most of the damage occurred in or near the towns of Bruton and East Bruton, about 48 miles (77 km) north of Pensacola, Florida.
"It kind of affected everybody", Jackson said. "But these mobile homes are built very close together, so they can cost far more than homes in remote locations".
Bynum reported from Savannah, Georgia.
In northern Florida, high winds - reaching up to 81 miles per hour - even flipped over an 18-wheeler truck, according to local media reports.
The storm also rained floods north of Lake Pontchartrain, Louisiana, and along the MS coast, flooding the streets and, in some areas, pushing water into homes. Later, the storm was drenching the Florida Panhandle and, well inland, a broad expanse of Alabama. The main threat of tornadoes is from Sunday afternoon through Monday morning.
Heavy rainfall totals up to 15 inches (38 centimeters) were expected across the southwest and western coastal areas of Mexico throughout the weekend. Forecasters warned of possible flash floods and landslides.