The storm's high winds have also been whipping up wildfires in Texas.
Lee has left more than a foot of rain in some areas, and was blamed for the death of a Mississippi man swept away by floodwaters, CBS News reports.
The storm is expected to dump between four and eight inches of rain as it crosses into Alabama. Tropical Storm Lee has already left 13 inches of rain in New Orleans and spawned tornadoes in Lilliana, CBS News says.
Jackson, Mississippi, received more than 10 inches of rain, and the city's mayor declared a state of emergency. The U.S. National Weather Service has issued flood warnings for the area.
While Lee's winds have weakened, forecasters are warning that slow-moving rain clouds pose a serious flooding threat to inland areas with hills or mountains in the next few days.
Flash flood watches and warnings are in effect across the southeast U.S., from the lower Mississippi Valley,to the Florida Panhandle and the southern Appalachians, CBS News reports.
Winds from Tropical Storm Lee have also whipped up wildfires in Texas, Reuters reports. Out-of-control wildfires have been tearing through parts of the state, destroying 300 homes near Austin and killing at least two people, and have been made worse by the high winds from Lee.
The outbreak led Texas Governor Rick Perry to pull out of a presidential forum Monday in South Carolina and return to Texas, according to the New York Daily News.