There are still spots available in Florida State Parks during that time but you may have to move a little more. The reservations will open up in the south end of Florida in April and May as the Snowbirds fly north.
You will fit in Jonathan Dickinson State Park - it has full hookup sites and wide open spaces - not much in tree cover or privacy though.
St. Augustine is a really nice place to visit - a lot of history - but you may not fit in Anastasia State Park - quite a few trees to dodge.
Check out the area around Ocala (fed and state park campgrounds). Lots of rivers with manatees in March though maybe not later in the spring. We stayed at Blue Springs (might be too tight for you) and saw dozens.
In May you should be able to get into Pennecamp State Park. It is at the top of the Keys and has FHU.
Your size will limit some of the state parks with the 'old florida' feel but you should be able to find something after March without many problems. Don't forget the panhandle, which gets busy starting in March (Fort Pickens is a great campground in which you should fit).