In order to file an appeal, he has to pay immediately. The money doesn't go to Carroll, it goes into an escrow account until she wins the appeal. But either way, Trump is out $83 million. No pay = no appeal.
If he doesn't appeal and doesn't pay, then in 30 days Carroll's lawyers will have access to his bank accounts and properties. At that point it won't matter if he writes a check, they can simply take what they are entitled to directly from his bank. If there isn't enough in his account, they can start an auction for his properties. The only thing Trump could do would be to declare bankruptcy, and this would only slow down the process not stop it.
Trump can get away with not paying his contractors, but he can't get away with not paying a court judgment. Of note, back when he owed Carroll $5 million, he paid on time.