Thanks for your question. Unfortunately you may not any rights here. Common Law marriage has been eliminated by statute. It is possible for the boyfriend to name you as beneficiary to life insurance or even bank accounts that would pass outside of probate.
He could also name you in a will as an heir. But under the law of intestacy(no will) you have no claim because of the common law situation. If he is able to do this before his demise then he can correct some of these things but otherwise I'm afraid that you are really limited here. The web page below gives the history of Common Law marriage in New York over the years.
My sympathy to your situation here. You may need to obtain a lawyer here. If the boyfriend is able I have done several bedside wills. It happens unfortunately as people procrastinate in thee matters...
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