Anastasia State Park is very nice. We stayed there January 7 - 11, during the record cold snap. The campground consists of five or six loops, all of which seem to have a variety of sizes and types of sites. Some are very private, surrounded with vegetation, and others are closer together, which would be ideal for camping with groups of friends. Each loop is almost like a little campground unto itself, except that bathhouses are shared by two loops. One thing to note is that the bathhouses do not have heat, and the high top portions are screened in, which can be quite brisk in cold weather! Taking a shower in 36 degree weather is an adventure!
However, the water is nice and hot! (Of course, you can use the TM shower, if you prefer.) The park personnel were very nice and helpful. We would definitely go back again. The beautiful beach area is a very short drive or slightly longer walk/bike ride from the campground, where there is a nice little camp store and boardwalk to the beach. The park is very convenient to the historic area of St. Augustine, and there are good restaurants and other conveniences nearby.
Other state park campgrounds we have used in Florida include Curry Hammock SP in the keys, Jonathan Dickinson SP on the east coast, Blue Springs SP between Orlando and Daytona, St. Andrews SP at Panama City Beach, and Big Lagoon SP near Pensacola. We had good experiences with all of them, and they each have their unique advantages and not many disadvantages. We'll be glad to share more details of our impressions of these parks if you're interested. We tend to like state parks better than the commercial RV parks because they are usually offer a more natural and quiet environment with less crowded sites.