Anastasia State Park is indeed very nice, with on-park access to the beach, but it can get very booked up, and iirc the campsites are very small (tho' that's based on recollection from quite a few years back when tent camping, tho' we did look at the RV sites).
We've stayed at the St Augustine KOA in the 5er, which is actually very close to Anastasia State Park. The park was nice and clean, with fairly roomy sites, and it has a small lake (with cabins on the far side).
It's about a mile from the beach (and pier), and you could even walk into the state park along the beach. (we actually drove into the state park after checking out, and parked there the next day while spending an extra couple of hours at the beach)
We did have a problem with a backed-up sewer connection, which they took a while to fix (but were very apologetic about afterwards, and did offer for us to move sites rather than wait)
A little bit further out is Faver Dykes state park, which we've also really enjoyed (stayed in our old pop-up, before getting the 5er). Very quiet park, away from everything, but be warned that the "roads" in the campground area are narrow and uneven (so you might struggle if you have a big rig).
Regardless of which campground you choose, in addition to visiting St Augustine itself I'd recommend visiting Fort Matanzas. You get a little boat across to the old fort (5 mins on the boat across the inlet) and they give some historical information, let you tour around the small fort, and may even fire the cannons. (There's also sometimes dolphins that'll swim alongside the boat). Good way to spend a few hours