Okay. I see a few options:
1) Ask for an exception to extend your stay in the dorms on the basis that you're waiting for an appointment. If you aren't offered an appointment, then the timer to vacate could restart. If you are offered an appointment, see if they'd let you stay in the dorms until you PCS to USAFA. Should only be a few more months. Be prepared to be told no, but it doesn't hurt to ask.
2) Try to find a friend who has an empty room off-base that they could rent to you short term. Be careful, make sure sure they understand and are comfortable with your situation. Definitely be prepared for what happens if you don't get accepted. Do you stay as a roommate? Do you find your own place?
3) Move-out and sign a lease for your own apartment. You should be able to break the lease with PCS orders to USAFA, should you be appointed. This would be more expensive and you'd want keep a very minimal lifestyle until you know the results.
I wouls ask about option 1 ASAP and be prepared to execute on option 2 or 3 while you're waiting to hear back.
P.S. As an officer you will be required to think critically through many problems. Take this time to practice weighing your options, their pros and cons.