Jim, how much time do you plan to stay down here? The Keys are cool, and that's where I expect to be doing most of my camping (on a rich friend's property on Upper Metacumbie!). But like John said, you gotta book early.
Even just making it as far South as Key Largo is very cool, and there's an Outdoor America and other camping sites there, right near John Pennekamp state park. You might have better luck getting a spot there. Also, after your long drive from up north, you might get a kick out of just unhitching the TD in the morning, and driving down to Key West for a day trip or overnight in a hotel.
It's a long haul but a NICE haul, a few hours from Largo, but it can easily be made into a cool day trip. DON'T make the trip down there on a Friday, and DON'T make the trip back on a Sunday. It's traffic-heavy with locals going down there for the weekends.
There are also some nice County Parks in South Florida which I like to consider "managed wildlife." It ain't the real thing, but they're real pretty, with lakes, watersport facilities and great toilets. In Broward County, Quiet Waters and Markham Park have some great facilities, and it's a quick drive to the beach. One is in Deerfield Beach, between Boca and Ft. Lauderdale, and the other in Sunrise, west of Ft. Lauderdale. They're for a pampered type pf camping though--where you can run out and pick up a pizza.
Tom S knows a lot about the southwest parks in the state. Check out his website for photos--www.tomswenson.com. It's the first trip he made in his TD.
There's a REALLY cool place right near Naples (east of it) called Everglades City/Chokoloskee. The latter is a tiny island connected to the first, and it's part of the Everglades National Park system, with glass bottom boats and more going out into the Ten Thousand Islands. It's as close as you can get to "old Florida," and an outstanding environment.
Here's a link, and look at the bottom of the page for two campgrounds.
http://www.florida-everglades.com/stay.htm
You can also stay in Faxahatchee Strand park near here, but no facilities. January, February--great time to be there. July, August--you're mosquito food.
Finally, you got kids? Depending on the length of your trip, maybe you want to spend a night camping at DisneyWorld's Ft. Wildernesss. It's AWESOME!