From what I understand, if your license says your home address or an address that is within 100 miles of your house, you must be listed on the policy and the price will go up drastically for a 19 year old.
If you go to Yeshiva out of town, you should get a license there, and revoke your NY license, and then you should have no problem.
Your best option is to call your insurance company and confirm all of these details. CH"V that an accident should occur where you will not be covered.