Dolo,
As hard as it might seem, a major pruning might solve all your problems. Pick one or two of the best looking branches and cut off all the rest. (Stick the cut branches in the ground or try to root them in some way.) Since the tree is actively growing, it will focus its energy on the branches you leave. If it tries to send up suckers from the base, nip them off. (It may not be great to prune when the tree is actively growing, so you could wait and do it in a few months...or others might reassure you that it is no big deal to do it now.) I think your Florida heat and sunshine can cure a lot of ills if you help the tree focus its energy.
I have to admit that I have not given any of my trees a major pruning yet, so I'm guilty of not practicing what I preach. But I realize it is a psychological block at this point and not doing it will soon start to do more harm than good.
Jim