We stayed at North Beach last year--would definitely recommend it. We liked the privacy each site afforded and the beach is just across the street with 2 restaurants within walking distance.
Another thing I would recommend is the Town Trolley if you are interested in historic downtown St Augustine. For about $35 you can use the trolley for 3 days--it has about 20 stops (hop off, hop on) and the tour guides will give you a lot of historical information.
The Fountain of Youth park is one of the trolley stops--the grounds are beautiful--lots of friendly little squirrels and peacocks (a few of the peacocks are solid white).