You should probably contact the park office and make sure that you will be permitted to use the site. I believe that you must have a handicapped placard or license plate to reserve a handicapped site in Florida state parks. If it's not an official handicapped site but is just listed as being ADA compliant then you should be fine.
Since I use a wheelchair we often use handicapped sites and we've asked about the rules at campgrounds across the country. What we've found is there are no across the board rules. Each state and even each park will handle it differently. Many parks will hold the sites until late in the day then, if they have not been taken by a handicapped person, anyone can take them. Other parks will not give them to anyone but a handicapped camper.