I'd bet the Gators were here naturally a couple hundred years ago, they were just "restocked" in the 1970's or 80's after being wiped out. It's hard for the young to survive the winters here, but the adults do fine. I'm guessing the one (or more) in Limestone spends the winter in one of the holes off the main channel. I've never seen it over the fall, winter and early spring when I fish the area, but there are numerous old pits and holes in the area. They prefer to eat something that will fit in their mouth whole and are attracted to splashing in the water. I've know guys in Georgia to wade fish arms length from Gators and have been doing it their whole life, but they also don't tie a string of fish to their waist, LOL. Much like snakes, wild dogs and hogs, stay aware and observant and give them their space and you won't have a problem.
There's no point going into the woods if all the animals are dead. You might as well sit at home on the couch and watch the TV.