There is also federal camping along US 41 in Big Cypress Nat'l Preserve - located in the Everglades and sharing borders with Everglades NP.
Midway Campgroundis really a pretty nice cg with an Everglades flavor. Also 'nearby', by Everglades standards, but within the Everglades is Florida Panther National Wildlife Refuge.
The little waterfront community where I live shares its borders with Everglades National Park, Big Cypress N. Preserve, Ten Thousand Islands Nat'l Wildlife Refuge (10,000 Islands run from Marco Is. on the west to Lostmans River in Everglades NP on the east - probably 100 miles), Collier Seminole State Park and the Gulf of Mex. Some "Old Florida" flavor and also decent seafood in either Goodland or Everglades City.
If you are just looking for overnight, zero hookups but next to a weir on the Fakahatchee Union River running between Lake O and the Gulf, I know of a virtually unused spot, with picnic tables, for 3 rigs. There is an iron box for payment and if I recall, it's $2.00 per person.
While it certainly is busier than summer time, even the week between Christmas and New Year will be before the height of the season which will begin in earnest after the holiday. That said, it will be busy. You will see that I 75 (Alligator Alley) and US 41 (Tamiami Trail) both traverse the Everglades. If you want to see anything, don't take I-75. If you are looking for gators in a even more natural habitat than, say, Shark Valley, keep your eyes open to the north ditch on US 41 on a sunny day for about a 10 miles stretch east of Rt 29 (Everglades City). An that same stretch of road, but on the south side, is a large habitat for the Roseatte Spoonbill. All along US 41 will be an abundance of birds - egrets, various species of herons, anhingas, osprey, eagles etc. Wading in Big Cypress is fun - not at all muddy or squishy