We have family in St. Augustine, so visit there often. Anastasia is our favorite, of course, but like you said - hard to get into during the "season".
I think it depends on how close to want to be to St. Augustine. We have stayed in Faver-Dykes SP, as well as Gamble-Rogers SP, and Tomoka SP, and never had any more problem with ticks or other bugs than anywhere else depending on the time of year.
We like to be right in St. Augustine Beach, so if Anastasia is full, we usually go to St. Augustine KOA right up the street. It is a little pricey, but very convenient and nice. An other possibility is North Shore Campground which is north of St. Augustine out on the beach.