I echo the sentiments of the others who said carrying a handgun for this is not very practical. Staying away from the water is the best policy. Any inland body of water in Florida may contain alligators. Assume they are there.
I am in a line of work that brings me into contact with them regularly. This often involves walking into the water in rubber boots or stooping at the edge to scoop up water. Entering their strike zone is part of the daily routine. Our company has guys doing this stuff all day every day all over the state and has for 30 years. Believe it or not, no one has ever been attacked by an alligator, and much more amazing to me, never been tagged by a snake either. We get way, way, way more exposure to it than just about anybody, so there's some perspective for you. A guy drowned once and there have been ATV and airboat accidents, but not one alligator grab.
There are various things I do to minimize the chances of an attack, but for the great majority of folks, staying clear of the water's edge is pretty failsafe. And keep anything important to you away too. I have been amazed with people tossing things into the water as part of a game of fetch with their dog, or with small children allowed to play in risky places. On a side note, I spotted over 20 cottonmouths in one lake within about 1 hour, and there appeared to be a nesting area with several just 6 or 7 lots down from where I saw several children running around barefoot in the backyard, near the water. I promptly stopped the boat and got their mother's attention and told her what I'd been seeing!
As a side note, people swimming in places other than the safety of a swimming pool take on another serious risk, which I call "mucking in". It is easy to get your feet caught in the right sedimentary soup and get yourself trapped and continue to sink. You had better believe there are times you cannot pull your feet free when it is halfway up a 15" or so boot. Imagine being in to your waist. You're not going anywhere. A construction worker here in Jacksonville died that way about a year ago. He thought it would be a good way to cool off on a really hot day.
Anyway, as alligators go, generally I have not found them to be aggressive toward me. Mostly they just want to get away. Yes, they do like to follow you around some times, but they keep their distance for the most part. I believe the following behavior is from curiousity, but I still always remind myself that it would be foolish to present them with any opportunities, because they may take one if it is given.
I have had a few that simply will not move and when I bring the boat around I corner them when I pass by. I have had occasions where I could reach out and touch some of those. None of those made a threat display when cornered. However, when on foot, I have had a few get testy and give me warnings. What they have done is get kinda close and make a loud splash, submerge, and pop up to watch me from a few feet farther out. Once this happened very close and behind me and a bit to the side, so that I barely caught it with the peripheral vision. That one could have had me had it wanted to. Although it wasn't a large gator, do not think that a smallish gator cannot beat you in its own element.
They are indeed very common on golf courses. I often treat golf course properties. Just recently I motored around the bend to the boat trailer and maybe 50 ft from the trailer I interrupted two of them trying to make more alligators.
I take them seriously. I do what I can to avoid confrontation. However, I am many times over more concerned with cottonmouths than I ever will be with alligators. Alligators generally move away and often make themselves known (if you get used to looking). A cottonmouth will, more often than not, hold its ground and let you walk right on top of it.
I heard one of the victims recently was a female jogger in her 20's, in the Gainesville area I believe. I can tell you that Lake Ashley at the University of Florida is chock full of them. Haven't seen all the details, but I wonder if it was around there.