I was going to recommend handgunlaw.us, but I've been beaten to that.
I was going to say I'm pretty sure you are good in all the gulf coast states with a Florida permit, which you've already determined.
What I will say, though, is it's not a bad idea to follow advice frequently put out by Gary Slider (who runs handgunlaw.us), and use the links on handgunlaw.us to check the licensing and regulatory agency websites for each state you will transit. Get your info straight from the horses' mouths, so to speak.
That protects you against recent changes to the rules that may not be reflected on handgunlaw.us, or against any possibility of misinterpretation or editing errors on handgunlaw.us
All that said, handgunlaw.us is a great planning tool. It just should not be viewed as a standalone tool.