I haven't lived in Florida since high school, so I am guessing here. But I honestly think you will be okay. If you are just traveling and sightseeing along with other tourists, I doubt you will be confronted, and all your documents are in order.
I suspect that the places where confrontation might be more prevalent are in bars. Alcohol fuels ignorance and tough-guy syndrome, so I would avoid them. Restaurants, truck stops (with showers), and hotel/motels, I would expect to be just fine.
Perhaps someone has a better perspective than I, but I believe you would be okay.