Sharks are way down the list of concerns while diving here, and the odds of running into a white shark are pretty slim. We're not sure what they're doing in FL waters; while there was one acoustic tag record back in 2015 of a white shark hanging around Palm Beach Inlet for about three months most of the satellite tag tracks have them booking it down the coast, into the Gulf, and then back around.
Whatever the case may be, I'm only aware of one instance off St. Augustine almost five years ago where a white shark made a remotely serious pass at a diver (bumped his tank from behind, kept going, then came back to circle a few times before getting poked in the jaw and bugging out). The majority of sightings down here, they just mosey on past and leave a bunch of shocked and ecstatic divers in their wake. I know the DM and one of the customers on The Wetter the Better who saw the one off Palm Beach the other week and even with Guadalupe Island under my belt I'm insanely jealous of them.
If there's a time of year to avoid sharks while diving down here, it's probably mid-fall ... which is when your dive trip is most likely to be scotched by a hurricane. Depends on the location too; Jupiter/West Palm for instance is pretty sharky, but to date aside from "cartilaginous catfish" (nurse sharks) I have to date only seen one juvenile Caribbean reef shark on the Key Largo reefs. Guess which way I go for diving?