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New Jersey recognizes domestic partnerships, so there is not a "full faith and credit clause" that you would run into in other states.
That said, it is unlikely that you would qualify for an annulment; an annulment is available when the domestic partnership was never legally entered or was otherwise fundamentally defective; for example, if one of the parties was not old enough to enter into it or if it was never consummated, that would be grounds for annulment. Instead, it would ordinarily need to be dissolved.
Hypothetically, if you were eligible for an annulment, you would just want to contact your local family law court house to request a dissolution packet.
Let me know if I may be of further assistance. Thank you.
The word that you are looking for is "dissolve". Asking how to dissolve a domestic partnership is kind of like asking how to fly an airplane; I can give a very general, vague overall description or I can get you the first step, but it is beyond the scope of this forum to walk you all the way through the process. I would still direct you to your local family law courthouse, but instead ask for a packet to dissolve your domestic partnership; most counties will give you basic information in their packet to work through the process. Obviously, having an attorney assist with the process is the easiest option.