For all state universitiesi in NY (the entire SUNY system):
"A person whose domicile has been in the State of New York for a period of at least one year immediately preceding the time of registration for any period of attendance shall be a New York resident for the purpose of determining the tuition rate payable for such period. All other persons shall be presumed to be out-of-State residents for such purpose, unless domiciliary status is demonstrated in accordance with guidelines adopted by the Chancellor or designee. (8 NYCRR §302.1(a)(6))"
You may prove this with a driver's license verifying an in-state address and/or a lease agreement. Tax returns are honestly the best way to prove residency, because states love it when you pay their taxes.
As for original date of driver's license, you need to contact the DMV. They might laugh themselves silly if you ask them for verification of original DL release. But in order to have gotten a New York License, you had to prove residency then . . . so how did you do it before?
Edit: I'm being dense--a couple of caveats.
If you went to an out-of-state school and filed taxes in that state for STATE purposes, you became a resident of that state and you may have to wait for your new tax returns to verify residency in New York.
If you went to an out-of-state school and didn't pay CITY OR STATE taxes, or DIDN'T register your dormitory address for federal, state, city or local taxation or voting, you're still a resident of New York. (Assuming you were one before you went to college)
If you're living with your parents IN NEW YORK and/or you remained a dependent on their tax returns, you are still a state resident of New York and you can use their tax returns to prove it.
If your parents are not residents of New York and they filed you as a dependent in wherever (Arizona), you will need a year-old lease agreement or New York driver's license to prove your eligibility for New York tuition.
EDIT: The military is a big exception to this: If either parent is military and had New York residency but is no longer stationed in New York, you as their dependent are possibly still a resident of New York. Same as if they are from Arizona but are stationed in New York: you are a resident of New York.
"Registration" in this context is school specific. It may either be the date of initial financial deposit for your space in class (highly unlikely) or the first date of the semester matriculation (HIGHLY LIKELY).