Yup, alligators do occasionally swim in the Gulf. There are alligators on Horn Island which is an island off the coast of Mississippi just two islands over from Dauphin Island. Going west from Dauphin Island the next island is Petit Bois Island, then Horn Island. For years one of the largest known alligators in Mississippi made his home on Horn Island. The national park service website mentions the alligators on Horn Island and reminds visitors they’re protected and to leave them alone.
See my post #34. Close encounter, yikes!