I live in Holiday, FL, about 30 miles north of St. Petersburg. I know (bicycle touring) Florida north of St. Pete like the back of my hand but have never bicycled south.
You could probably go to, say, Sarasota and on down the Gulf coast to the Tamiami Trail (US41), across the state and on down to Key West. There are intermitent trails alongside the Overseas Highway. The Sunshine Skyway (I-275) south out of St. Pete is closed to cyclists. This is a trip I have had in the back of my mind for years. Unless you have a tolerance for heat you should do it in the dead of winter. I have lived in Florida since 1976 without air conditioning so am well acclimated but northerners often have a hard time here.
The west coast is, frankly, a lot more scenic and interesting than Florida. If you could do a trip there between Labor Day and mid-October you should have a fine, if physically demanding, tour. This would avoid the summer crowds and the winter storms.