I do not think this information is correct. In late 2009, the Military Spouse Residency Relief Act became law, which allows a military spouse to pay taxes in their state of domicile. There are some conditions; namely, the spouse has to be living in their new location for the sole purpose of following the active duty servicemember. The spouse can still choose to pay taxes in the new location, if it's advantageous to do so.
On a separate note, there is some confusing terminology that needs to be cleared up. "Residency" is a nebulous term without a clear legal definition; it really shouldn't be used. The proper term is "domicile", and even then there isn't a universally recognized standard for how to establish domicile. For a servicemember, it's best defined as the state in which you intend to establish your household after you end your active duty service. Proving or disproving intent is extremely difficult, but things that you can do to indicate your intentions include maintaining a driver's license in a state, registering your vehicles there, executing a last will and testament in a certain state, or owning property. The single-most important factor in determining domicile is where you vote.
Your home of record is simply the location from which you first entered active duty. For practical purposes, an officer's home of record is permanent and cannot be changed. One's domicile may change many times, but your home of record is always the same. This only really has consequences when you are separating from the Army.
Check your orders and additional paperwork to see where the Army thinks your home of record is. If you would like it to be Florida and they're listing your medical school location, then you may want to have it changed. And that's something you definitely want to take care of now. You will probably have to convince them that you never really established a household in your current state, which may be difficult or impossible to do.
For tax purposes, if the Army is withholding money to a state other than Florida, then you'll need to visit the finance office to have that fixed. It's a relatively simple form. If you've established your domicile in Florida by way of voter registration, car registration, etc., then you should be on solid legal footing if another state were to ever come looking for their chunk of your pay.