I live and work in Albany NY. The company I work for is incorporated in New Jersey. They had been withholding NYS SUI and DBL for me until this year. Now they say since they are a NJ company they must withhold NJ SUI and DBL and not NYS. Question (1) Is this true and if so what happens to my NYS status if something happens (2) Does this mean I would not be covered under NYS unemployment insurance anymore if I was to lose my job (am I covered under any unemployment insurance) (3) Am I still covered under NYS Disability if something were to happen to me (and am I covered under any disability at all in this siutuation) since I am a NYS resident?
Thank you for any assistance you can give me in this matter.

If you live and work in NY, your income is not subject to NJ taxes even if it is paid by NJ employer.
Read: Your U.S. Tax Return: Working or Living in Two or More states