Well that time period is the busy period aka the season in southern Florida. Meaning there are many snowbirds down there. The main Everglades section is the Flamingo entrance which is near Homestead FL. All the south in other words. Hwy 41 to the north leads past the other ENP entrances. Shark Valley which is worth a trip to take the tram tour, and the Everglades City entrance to the west which is where you can take boat tours. The Big Cypress Swamp is also located in that same area. Collier-Seminole State Park would be the best place to camp for those sections if it has any vacancies. For the Flamingo area there is a big CG all the way south in the park right on Florida Bay. I don't think it has any services and may be full. I have never seen any commercial CG outside the Flamingo entrance but there could be some there.
Camping in the Keys without a reservation could be very difficult that time of year, and people start making reservations more than a year early.
Frankly I don't see spending more than a couple of days at Flamingo and the same on the Hwy 41 areas. If you are a bird watcher then more time could be called for.