The area you're going to be in is probably the best. Check out Corkscrew Sanctuary, seen indigos there more than a couple of times; Big Cypress Nat'l Preserve (check out Loop Road off of HWY 41, and Turner River Road, also on 41). There will be some dirt roads you can drive on to go a little further in.

Also see Fakahatchee Strand, especially the entrance off of SR29, about 14 miles east of Everglades City. Janes Scenic Drive is really cool, and you can take any car in there, but it will get pretty bumpy at times. After 11 miles, if it's anything like what it used to be - I haven't been in a few years now - there will be some paved roads. But make sure you know your way back, they are all unmarked. I've seen some snakes out there as well.

Finally, go out to Everglades National Park, way south of everything (the entrance that is past the end of the turnpike, not the one near Everglades City). You'll likely spot a Burm out there; I've seen a few over the years. But that is the most touristy of all the spots I've mentioned. Good luck.