Without reservations this time of year, finding a camp site at the state and federal parks can be a problem. But, most of the state parks hold at least one site as a walk-in, which means that it cannot be reserved. But, you need to be lucky and be there when the gates open in the morning and the current camper has to be leaving. Once you occupy a walk-in you can stay for 14 days.
The campgrounds nearer the coasts are more difficult to get in. Inland is easier, but those are often more distant from grocery stores, etc.. We've camped at Gold Head SP in Keystone Heights (about 30 miles east of Gainesville) and there have always been open sites when we were there.
Salt Springs Recreation Area in the Ocala National Forest is a federal park with full hookups. They divide the trailer sites, with 1/2 reserveable and 1/2 non-reserveable. So, there's a chance that you can find an walk-in there, but if you don't, there's always dry camping
in the forest nearby.
A note on the Florida weather. 40 - 50 degrees F can feel pretty chilly down here. The dampness reaches right to the bones. We try to get as far south in the state as possible during January, then gradually work our way back north as the weather warms up.
Good luck and safe travels