For your ACT score of 34, was the writing section offered? That seems to be key for Princeton: Princeton University | Standardized Tests
"A complete application includes official scores of the SAT Reasoning Test or ACT (with Writing, where offered), and two SAT Subject Tests."
For the date of your 34 ACT, if the test was offered with writing, but you opted not to take it, then your test would be invalid for admission purposes. However, if the test was offered without an option for writing, then it might be valid.
As others have said, it's best to call the Admissions Office and ask. The number is (609) 258-3060.