While an unplanned fuel stop isn't common (I've had none in my 185 flights in the last few years), it's routine enough that everyone knows how to handle it, and the best way is by keeping it simple: stop, refuel, and get going. No passenger shuffling, no baggage changes (the holds are last-in-first-out, so they would have had to unload all the bags going to FLL to get the ones going to JAX), and all-around fewer chances for something to get messed up.