I have done tons of back country camping in the Everglades, ever since I was a kid. You're going at the right time, December/Jan you won't have to worry about the heat, and more importantly, NO MOSQUITOES. Those are what make camping in the Everglades in the summer almost unbearable, heat is one thing, swarms of mosquitoes is another. In fact, just the summer in the Everglades is full of all kinds of insects, it's not nearly as much fun as in the cooler months.
There are of course still snakes/alligators, but depending on where you camp really the only thing you would have to worry about is snakes. Alligator attacks are very rare, and unless you're taking midnight swims in some remote lake you won't have anything to worry about.
Love the Everglades, it's one of the most remote places I've ever camped (there's a difference between looking all around and seeing nothing but forests and mountains and looking all around and seeing nothing but sawgrass). It's almost similar to desert camping in the style, minus the fact that the entire Everglades is basically just one big low slow moving river.