Yes, if you are becoming a resident of New York, even if just for a year or two, you will need to register your car in NYS as well as get New York car insurance and driver's license.
The New York State Department of Motor Vehicles states that you must apply for a NYS driver license not more than 30 days after you become a NYS resident. You also have 30 days after you become a resident to register your vehicle in NYS.
When you apply for a New York title and registration you must have valid NYS liability insurance. You cannot use proof of insurance from another state under any circumstances according to the NYS DMV.
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