New Orleans - you said food and art. Not enough can be said about this amazing city, and it is very romantic. And fun! But a lot of the fun is not for new brides in maternity clothing. You could visit Fairhope on the way.
Apalachicola - very quiet and tiny, for seafood and nature lovers. And isn't loving nature really all you need on a honeymoon (depending on your due date of course)? Lots of places around to explore, but you'd never have to leave the Gibson Inn and the fine Avenue Sea Restaurant, except you'd have to try some of the other great restaurants in town. Tamara's Cafe across the street and Cafe Con Leche on the opposite corner owned by the same family are worth the trip. They have a chocolate shop / confectionary attached to the cafe which will blow your mind. Unfortunately the main attraction, the raw oyster, is off limits to expectant moms.
Amelia Island/Fernandina - This area might be your best bet because of all the diversity. We like staying downtown Fernandina Beach by the river and enjoying the incredible restaurants (Espana, Bonito, Les Clos), lots of shopping also. So many great places to eat and shop in a compact, walkable, old downtown. There are several awesome B&Bs and small Inns. Nearby Amelia Island resort out on the beach offers luxury if you prefer. That way you can have the best of both worlds. You can take a day trip to Cumberland Island. Google it and you'll be sold, but not a good place if you go into labor. And then you could always visit or stay in Jacksonville or St. Augustine. They would be better choices to stay in if you prefer to be in the middle of the action. The Columbia Restaurant in St. Aug is the highlight of any trip there.