There's not a lot of National Parks in Florida. Most of them probably don't have the camping facilities you need. The few that do are in a swamp

Two(Biscayne, Dry Tortugas) have awesome diving(and have camping), but are boat access only. There are a ton of State Parks. There's also a variety of diving available, from spring to ocean. If you want to do both, probably a better location would be somewhere in the middle of the state, that way you're within an hour of ocean and springs, maybe like Silver River or Wekiwa Springs. Blue Springs has scuba, but is also a manatee preserve, so you can't scuba the springs during winter. If you want beachfront, Gamble Rogers/Flager Beach would be good. Although I would imagine that all the beachfront parks would be harder to get a host position. You could do North Florida in the spring and move south to the keys during the summer when it's the slow season and most RVers will have left. Although by then they might not need a host. Summer will be hotter further south, but the water will be calmer. North Florida waters will be cooler in the springtime, probably in the 60s to low 70s, the springs are generally constant at 72. You could also do the Destin or Panama City area. If you end up in the Keys, having a boat would be really good since there's no beach diving to speak of.
Here's the list of Florida state parks that have scuba listed as an activity within the park. A lot of them are day use or boat access only parks.
Welcome to Florida State Parks
This book would also be helpful:
Amazon.com: Diving Guide to Underwater Florida, 11th Edition (9781878348395): Ned DeLoach: Books
I don't know about county parks other than that there are 67 counties and you'd have to search each individual county for what their parks offer. The beaches are public accessible, so if you have the book, it'll tell you where the diving is good and whether you can do beach dives or need boats. I do know a couple of parks in the 2 counties I frequent that have camping don't have RV sites. I'm mainly a tent camper, so that's not generally an issue.