As previously mentioned, Flounder's, in the core, on Pensacola Beach (not in Gulf Breeze). Very big servings and generally good. They overlook The Sound out back.
Peg Leg Pete's, about 1 mile west of the core down Ft. Pickens Road, is also very popular. The wait can be long, especially on weekends and most any night. Get there around 11, when they open, and you can usually be seated right away. Or eat a late lunch/early dinner during "off" hours. Great oysters, including raw ones. Across the street from the Gulf. Backs up to a marina on a canal which leads to The Sound. The owner also owns Sidelines Sports Bar and Restaurant (in the core area) and Maria's Seafood (in Pensacola) so his stuff is fresh and genuine. I have to have their grouper nuggets on a regular basis or I go into withdrawal.
In Gulf Breeze - the little city between the 2 bridges - is Aegean Breeze. Exceptional food, not all of it with a Greek flair. Very fresh seafood. Fish, shrimp, calamari, etc. I'm not a gumbo person, but their gumbalaya is said to be excellent by those who've had it. I love their shrimp Creole. Their pork chop dinner is a bargain and delicious if you want a change. They're closed on Sunday.
At Joe Patti's fish market, on Main Street in Pensacola, you can get both raw and some cooked fresh seafood. Worth a trip just to see it in operation. They have service by number down to an art. If the Royal Red shrimp are available, try some. A must-have for me when I have company coming.
Farther east on Main Street, back toward the 3 Mile bridge which takes you to Gulf Breeze and the bridge to the beach, is a restaurant complex. Fish House, Atlas Oyster and The Deck. Overlooks a small marina on the bay. Nice selection of seafood - and non-seafood items. Go for lunch and the prices are better and the portions are still ample. The Grits a YaYa are one of their signature dishes and extremely tasty, IMHO.
Patti's and Fish House/Atlas/Deck are the closest ones to the Naval Aviation Museum that I know about. I'm sure there are others, but since I live on The Island, I rarely get to the west side of Pensacola. In fact, I rarely get any farther than Gulf Breeze if I can help it.
And if you have the time, you really should go to Fort Pickens in the National Seashore on the west end of Santa Rosa Island. Well worth the time.
If you go to the museum during their show season, the Blue Angels practice on Tuesdays and Wednesdays. After the Wednesday sessions they go to the museum and sign autographs and chat with the crowd. Also fun to watch them practice from Fort Pickens.
Hope this helps a little.