FOF museum was open in April 2016; we toured it. The original building, that basically had a bunch of flight sims, etc was not. We stopped in there about 15 yrs ago & didn't even go in, because we weren't interested in the sims & there wasn't much of a museum then. You cannot fly in.
LuLu's (Jimmy Buffet's sister) Restaurant in Gulf Shores AL is walking distance from the airport, or you can borrow a crew car from the FBO.
I've heard good things about Lambert's but haven't eaten there. Some friends flew down several years ago, and they picked them up at the airport, took them in the back entrance past a very long line of waiting customers & give them immediate seating (no waiting).
There's a runway (no services, last time I visited) on Dauphin Island, just south of Mobile AL. Some of the restaurants on the island will pick you up, but the nearest one is about a 15 minute walk.
Big +1 for Pensacola NAS Museum; only problem is you need to allocate at least 3-4 hours for a quick run-through. I've been there several times, and my wife and I are planning to go again next month, on the way home from a family trip to Ditzy World. Word is that they've opened another building or two full of a/c.
Charlie