When we go to St Augustine, 30 miles, we stay at the St Augustine Beach KOA. Its well positioned near the beach, the Amphitheater , the A1A Sunset Grill and the Gypsy Cab Company Restaurant. Stops we often make. It is also near A1A south which is also a prime kayak route to Matanzas for us.
https://koa.com/campgrounds/st-augustine-beach/
Some folks I know like the Stagecoach up on hwy 16 near the outlet mall.
https://www.stagecoachrv.net/
Some seem to like North Beach RV camp. It is close to beach one side and ICW the other side. It also is near a couple of good restaurants. But they are REAL proud of their sites. Even for me its a little much.
https://northbeachcamp.com/
The St Johns RV park and Storage is at the flea market and often has sites. It is more of an overnight stop.
http://www.stjohnsrvpark.com/index.php
April and May is over for the "seasonals" but St Augustine tourist season goes all year long.