The Florida summers are VERY different from California. Used to store mine in a storage lot and the ones left uncovered are not nice to look at.
Yes - cover it by all means. If you properly cover the sharp edges and use the straps to secure it, you won't have a problem with wind damage. I'm sure it won't protect it in a Hurricane, but that kind of damage is what ionsurance is for.
I use an ADCO cover with a white top. Temps inside the covered TT are noticeably lower when it is covered.
If you put a good wax job on it before storing it, it will look like it did when you covered it upon uncovering.
I never experienced mold issues during storage.