Congrats on going to NYU! My son is also anaphylactic to peanuts and tree nuts. You should reach out to NYU dining services, and also check out FARE (Foot Allergy Research and Education) website. They have a college list of some schools and the services they provide.
It’s really scary, but if you have your emergency protocol, meds, and a discussion with food services, I think you will feel better. Also bring some same foods with you. Also make sure to communicate with your roommate what you need. Best of luck and have a great time in NYC!