There are alligators in Gulf State Park's lakes and connecting waterways, including here. This is not a rumour used to keep children in line, and it is not a reason to avoid the lake.
What it is, is a set of habits worth having.
Do not swim at dawn or dusk, which is when alligators feed. Stay in the open, used swimming areas rather than exploring the reeded margins. Keep dogs out of the lake and away from the bank entirely, because a dog in the shallows reads to an alligator exactly the way prey does. Never throw food, fish scraps or bait into the water near a bank where people swim. And if an alligator is visible, get out and move along the shore rather than watching from ten feet away for a photograph.
Feeding an alligator is illegal in Alabama, and the reason the law exists is that a fed alligator stops avoiding people. Almost every dangerous animal on this coast was made dangerous by somebody with a bag of chicken.
Snakes, including cottonmouths, also live around the margins. Stay on the mown and paved areas and you will almost certainly never see one. And bring repellent: the lake edge at dusk between May and September belongs to the mosquitoes.