You cannot consider yourself a resident without establishing residency in whichever State you choose. However, most if not all States allow for non-resident hunting. Residents generally get first choice in lottery permit drawings. And, non-resident fees are generally more than resident fees.
I recommend studying the NY regulations first and foremost. You may find an outfitter who will guide you and provide land access.
Going be time consuming and potentially costly. If you are serious, it will be well worth it. Nothing much more exciting than calling a gobbling male turkey into range and successfully harvesting it.