I don't know the Tampa/St. Pete/Clearwater area too well for camping (most of my experience down that way is with skating), but a resource I recommend is
Florida Hikes! It seems to do fairly well at covering an areas hiking resources (just look for "backpacking" options, and you'll be golden).
Another resource that might be worth looking at is the
Florida Trail website. It doesn't pass too close to the Tampa area, but if you're driving, that shouldn't bee too much of an issue.
Generally, hammocking is okay in wildlife management areas and water management areas, as well as state forests and national forests, but not state parks (the price of tourism, I suppose). However, it's always best to call ahead to the local land management before going, both because it's hunting season in most of Florida through until March-ish and because they should be able to give you an hard answer on the hammocks being okay.
I hope you enjoy your stay down here! Feel free to PM me if you come up Gainesville way.
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