I am the co-signer on a car loan in NYS can I register the vehicle

How is title held? Being a co-signer does NOT give you ownership.

It is still held by the lien holder

I am not sure but the bank told me I could just go to DMV and have the vehicle registered in my name. The person I signed for no longer wants the vehicle and if I am going to be resposible for the loan I would much rather trade the vehicle in for a newer vehicle.

The lien holder is not the owner. The owner has to sign over title to you subject to the lien holder's agreement. Then you can trad it in.

... I would much rather trade the vehicle in for a newer vehicle.

Then do that. Bring the vehicle to the dealer and let the dealer take care of the registration issues. You will want your new trade-in registered in your name, not the one you co-signed on. Dealers do that all the time, so they should know how to handle the paperwork.

Then do that. Bring the vehicle to the dealer and let the dealer take care of the registration issues. You will want your new trade-in registered in your name, not the one you co-signed on. Dealers do that all the time, so they should know how to handle the paperwork.

The owner f record will still need to sign over the title back to the dealer.