You are playing with fire here.
The oral contract is binding because it can be completed within one year. A longer contract would have to be in writing to satisfy the statute of frauds.
However, the program, that you orally agreed to work for, would have to prove damages. A program must mitigate its damages by finding a suitable replacement. Given that these spots generally are competitive and that your talents are not unique, the program could easily find a replacement for you to mitigate its damages. Hence it suffered nominal damages. Suing for nominal damages will cost more in expenses than the actual recovery.
The real problem is one of integrity. You accepted at one program and are now speculating on that program's dime. That program is bound to you.
They have a right to call your first choice and tell them what happened. That other program now has grounds to rescind its contract with you for cause; i.e. a lapse in integrity.
You should be open with the program that you orally agreed to. Tell them that you really want to go to your first choice. They will probably let you out of the contract. This web of deception that you are weaving may get you caught in a very sticky situation that could leave you with no spot at all.