The Law:
"You must exchange your-out-state driver license (from another U.S. state or territory, federal district or Canadian province) within 30 days of becoming a resident of New York."
https://dmv.ny.gov/driver-license/exchange-out-state-driver-license
Legally you should have transfered your license to NY and got a NY license card 2 months ago. The license that says you're under 21 is no longer a valid license. You need to get the one from your mom's place and get your license transferred to NY as soon as possible.
Don't worry about being late with transferring it too much. Police can fine you but they usually don't.
Don't get pulled over with that invalid license that says you're under 21. Not being pulled over in certain areas of NY is impossible. Depending what area you're in you're going to get pulled over at some point. Get this corrected so you're legal as soon as possible.
You're not legal both driving and living there right now.