Should I admit to cheating when it appears I cheated but I did not?
I'm panicking right now, as I was supposed to be studying for finals


If you did not cheat, you do not need to be panicking.  

I recommend these steps:


Tell the professor that you did not cheat.
Tell them that right now you need to focus on studying for your exams.
Tell them that you look forward to straightening the matter out after the exams.
Request to take the exam at the normal time.  Point out that the professor can always require you to take the in-person exam later, so it is possible to resolve the accusation later.
Go back to studying.


After you have taken your exams, then learn and follow your university's procedure for resolving these disputes.  Keep it calm and logical.

If you are still feeling panicky when you need to study, talk to your university's counseling service.