And if my license is no longer valid and I only have a learner permit, I can't drive without restrictions anymore?
Exactly. NY State law requires a supervising driver to ride with you. The supervising driver must be age 21 or over and must have a valid driver license to operate the vehicle that you drive.
What a bummer, but no can do. So what if my visitor status is about to get extended?
Then you must bring a completed application for permit and a driver license to a DMV office.