On the east end of Alligator Alley, just off I-75, there's a nice little campground in Markham Park, a public campground owned and managed by Broward County Parks and Recreation. It's $40 per night for non-residents ($30 for residents).
Here's a link: Markham Park
A little bit farther south on I-75, just before you exit for the Turnpike to the Keys, is C.B. Smith Park, another Broward County park with a campground. Exit I-75 at Pines Blvd.
Here's a link to that campground: C.B. Smith Park
Those two parks look like they were made to order for your trip.