In Alabama, the sling is legal for all non-migratory game birds except Turkey. It is also legal for varmints and small game animals that
are not a raccoon, opossum, fox, coyote, or feral pig.
What does that leave? Mourning Doves, Nutria, Rabbits, and the like.
The sling is legal by non-exclusion. The only prohibited weapons are centerfire ammo guns, rimfire ammo guns larger than .22, shotguns above 10 gauge, shot larger than No. 4, and scoped pistols. Even blow darts are specifically mentioned as legal, as well as the
sling shot which has an ambiguous definition in Alabama law. Have at, but tread carefully.
I do not have the accuracy to hunt, so I have not tried. You'd be better off not hunting either, at least for the immediate future.