By "cabin", do you mean woods? Honestly, not that many people other than hunters like to spend time in the woods in Florida -- there isn't much there other than snakes, mosquitos, palmettos and pine needles. Fall is hunting season, though, but I understand that licenses for out-of-state hunters are quite pricey.
If by cabin you just mean any vacation where you stay in a self-contained house and do all your own cooking, then sure, there are a zillion little beachy places that fit that description. Choice of beach in fall means choosing your dates carefully; September is still full-on hurricane season, but by October the water in most of the state is getting a bit cold for swimming.