You have a feeling but then you asked us. If I were you, I wouldn’t assume - and I’d ask them (the old school). That, or you get a graded 1-2 page paper this year….this is your choice.
btw - records don’t just disappear.
Princeton has a requirement - you can meet it or not. That is up to you. You may have to submit the paper you don’t want. You might get a C on it…who knows.
But I suspect applying with a paper, even if it’s not the one you want, is better that not.
And again, you are making assumptions about your old school. How about reaching out to them?
A student was panicked earlier today about college classes they took - they said the transcript was unavailable. Guess what - they were wrong.
I’m guessing you are wrong too.
Good luck.