Gulf Shores has an excellent pool with a seperate area for young children. You can camp away from the water if the thought of gators worries you. You would want to drive to the beach anyways since you will probably have quite a bit of things to bring to the beach. While you can't reserve individual campsites, you can ask for an area. I would call the park to reserve rather than do it on-line so they can suggest the best area to meet your needs.
Henderson State Park is a beautiful campground, one of our favorites. It is a walk to the beach across a boardwalk. No pool and we saw they removed the small playground area they had for kids near the campground when we visited in November.
We like both campgrounds but with kids would pick Gulf State Park.