Every '51-'52 Plymouth I ever had came from New York and the old registrations had the motor number on them, not the VIN on the doorpost.
I would recommend contacting the NYS Department of Motor Vehicles (their website is www.nysdmv.org) They are very easy to deal with and will help you to correct the paperwork.
In my case, I had a 1974 NYS registration for my '51 business coupe, with the engine number on it. When I moved to New Jersey, I decided to get a NJ title for the car. When I went to New Jersey DMV, they refused to issue me a title with a VIN on it. I then contacted NYS DMV, and they told me to bring my old paperwork, a valid insurance card showing the VIN, and a tracing of the VIN off my car. They issued me a New York State registration with the VIN, which I then turned-in to NJ to get a title here.
New York also has a procedure for lost or insufficient paperwork. Call them, explain your situation completely and honestly, and they can probably help you.