There must be thousands of campgrounds in Florida. Try to get a copy of the magazine-size directory of Florida State Parks, and another similar one listing many commercial campgrounds. I've found them both at the state welcome center on I-95. The SP guide has full descriptions of all the parks by region and you can circle the ones with campgrounds or cabins. Water & electric hookups are often available.
Also there's Everglades National Park, which is great to visit in the winter and spring, with electric sites at the Flamingo Campground.
Most activities, ranger kids' programs, socials, etc at all types campgrounds will be underway full strength in the winter season, but outdoor water sports like swimming are fun in the summer. (Bring bug repellent.) There are some SP with springs and several on the Gulf beaches up at the Panhandle.
For St. Augustine, book Anastasia State Park, which has a beach. There's a string of Other SPs down the Atlantic Coast to try. For example, Jonathan Dickinson has boat trips back into the Loxahatchee River. In the Keys, John Pennekamp SP has a small beach and a couple of easy hikes. Bahia Honda SP has a larger beach, and has some small camping sites not available for RVs that might work for you.
You may have an easier time getting tent sites than RV sites in popular state parks like Long Key, so enjoy that. However, if you are camping in the summer, my guess is a tent won't be cool and comfy.
