Years past, I've lived near St. Augustine. Now, domicile in Green Cove Springs as a full timer. Can't help with camping other than Anastasia SP. With any kind of time to explore, I'd skip the tour company. Pick up a paper guide and take in what interests you; whether it's the Old Fort, Fountain of Youth, Ripley's, or the Alligator Farm along with others. Be sure to dine at O'Steen's, 200 Blk of Anastasia Blvd. Looks like a small dive from the outside. But, it's where the locals go for shrimp.
I live in St. Augustine. Ocean grove is nice. It is on the West side of A1A, a couple of blocks from the ocean. There are also Bryn Mar and Peppertree in that area. Another good choice would be North Beach campground, which is a couple of miles North of St. Augustine on Vilano Beach. We actually go there a couple of times a year for family outings. It is on the intercoastal waterway and right across the highway from the beaach (the island is narrow there). I would choose it over the others.
North Beach is great! Both ocean AND ICW beaches - lots of casual walking area within the park - parking and access to the uncrowded ocean beach. The only downside is you don't have the same rights to drive on the (St AUg.) beach. Vilano has a nice beach to drive on (4WD, if you're smart),but it lacks the same feel. Driving ST Aug beach (any car) and parking for lunch or beach activities is great!