Having gone to FL for the past 5 winters, I have found that space is very limited for any long stays at any of the parks. We generally stay south of tampa at state parks, but they have a 14 night limit. At this late date you would be hard pressed to find anything longer then 3 or 4 nights at a time at an open spot. FL state parks are usually around $30/night give or take. Get to that reservation system thru reserveamerica.com
The panhandle parks, while cool in Jan, and Feb are probably your best bet for finding longer stays. The nice thing about them is they have some of the nicest beaches that are nearly deserted in the winter. St. Joseph Peninsula SP and St. George Island SP are two of my favorites. The sun does shine in the panhandle, but it also gets a Canadian clipper air mass for a few days and temps get into the 30's at night and 40-50 during the day.
Getting close to the gulf side beaches I would think would be nearly impossible, as those spots are booked solid 11 months in advance. Looking for something inland, say an hour or so from the coast would be easier to find. Something like Highland Hammock SP in Sebring, FL.
Some National campgrounds administered by Corps of Engineers or National Forest or National Parks are available, but about 1/2 of those have no electric. The ones that I have been to is Midway Campground (electric) in the Everglades, and W D Franklin (electric) near Ft Myers, (Alva). Get to the reservation search through recreation.gov... take a genny and your choices open further.
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