I have a simillair issue here, one of my dogs went after one-got it, and managed to get bitten, but not badly, thank goodness.
I havent had to shoot any since that incident, though. But, I do keep my H&R .410 handy when I let the dogs out at night -both for use vs. coon, if needed, as well as for snakes (mostly).
Frankly, I think you could accomplish what you need with just a BB gun, really. Just step out before you take the dogs out, send a couple of BB's at, or near them, and they will scamper off pretty quick.
here's the florida regs on Coon "hunting" -it doesnt specify doing so on your own property though...
Raccoon, opossum, coyote, beaver, skunk, nutria
Statewide
Year-round
by all legal rifles, shotguns, muzzleloaders, crossbows, bows and pistols (D)(F)
Hunting raccoons or opossums at night is allowed, but only .22*-caliber rimfire firearms (other than .22-mag*nums) or single-shot .410-gauge shotguns (using shot not larger than size 6) may be used. Firearms may not be loaded until immediately prior to shooting raccoons or opossums. Hunting raccoons or opossums by displaying or using lights from moving vehicles, vessels or animals is prohibited. It is illegal to transport wild-trapped live raccoons within, into or from the state, except by FWC permit or as authorized under existing rules.
Bag Limit: No limits on any species