You should not register to vote in Florida, and you'll need to replace your Florida driver's license ASAP. It's associated with an address that is no longer valid, and most states make you relicense (and reregister your car) within a span of time after moving. New Jersey gives you 60 days, Rhode Island gives you 30.
Beyond that, it comes down a question of where you're going to consider your permanent residence. You should both get your license from that state and register to vote there. NJ requires that you have been a resident of the county for at least 30 days prior to the election, RI requires simply being a resident.
Presuming you're qualified to register in both states, you can choose to register tactically - currently, both states are projected to vote for Clinton in the general election, but it's more of a sure thing in RI than NJ. However, keep in mind that this also affects more than just voting: State taxes, possibly in-state vs out-of-state tuition, etc. You're choosing which state is your official permanent residence (until you next move).
All that said, should you prefer RI or NJ? That's a decision many college students face when they go to college out-of-state. It will depend on how often you go home (frequently, summers, almost never...), whether you have your own car/insurance or you're on your parents', and so on. There's probably other stacks which can provide better advice on that front. (Money.SE or Acadamia.SE are the two that come to mind, but check their tours and guidelines first.)