Hi AF,
Sounds like you just might like the 8th AF museum.
As you know St. Augustine isn't far from Jax so you could stay there. Not familiar with the campgrounds in the area but I'm sure someone is.
Traveler's World is very close to downtown and you can easily drive to the Alamo area, and park in a parkiing garage, or take the city bus that stops right across the street. I prefer to drive so I don't have to be dependent on the bus schedule. Also within a mile or two of the campground are 4 or 5 Missions which are easy to get to from the campground. We enjoyed that very much. Met a Monk at one of them and had a great conversation about his life. Funny guy with some neat stories.
On your way out of San Antonio you might want to swing by Fredricksburg, TX. They have the Admiral Nimitz museum and the George Bush Pacific War museum. Both are very interesting.
If you don't mind boondocking for a night or two a stop at Steins, NM can be entertaining. It's an abandoned railroad town that a couple has purchased and is preserving the history with what's left. The wife is a very good guide and will take you around the small town for $2.50. You can stay the night but only if you like trains. There are supposed to be 25 trains in 24 hours. I love them so they don't bother me. You are about 100 yards from teh tracks. There's a dirt road crossing so they have to blow their whistle when coming through.
It was a water stop for steam engines and became a ghost town overnight once they started to run diesels.
If you plan on moving frequently you might want to look into Passport America. It's a group of campgrounds that provide sites for half price. Cost is around $45/year or so. We make that back in three nights lodging! They are not always the best campgrounds but we have found them more than adequate for a one night stop and, sometimes, even longer. One in Benson, AZ is a resort which would have cost us $32/night if we weren't PA members.