I have always liked Dauphin Island at the mouth of Mobile Bay. Pretty nice city owned campground, with full hook ups and shade, a revolutionary war fort, an aquarium, great fresh seafood, museums and ships. Nice nature preserve and aviary.
The campground caters to fishermen, mostly. It is essentially a great big fish camp, but a nice one.
Speaking of fish camps:
http://www.felixsfishcamp.com/
Likely the best place to eat on the Alabama shore. Very near the USS Alabama.