Big eyeroll on the "kidnapping" threat. In many states, in the absence of a custody agreement, the mother is presumed to be the sole custodial parent even if there is a father listed on the birth certificate. New York does not appear to be one of those states, at least with information that is easily accessible on the web.
He can't have you arrested for kidnapping because there is no document that grants him parental rights, and the police have better things to do with their time than enforce non-existent court orders.
That said, he can certainly file for custody and/or visitation and drag you back to NYS for hearings at later dates. Of course you would also then file for child support.
If you can't afford an attorney (most single moms can't, I never could) you can call up the clerk's office of your local family court. Many courts have a free "lawyer of the day" service and you can meet with someone who will be able to answer your questions. I have found that the clerks themselves also had a lot of reliable information. We have successfully obtained custody of both my son and my husband's daughter without attorneys, so it can be done but I would try to speak with someone who can give you a clear answer on the presumption of custody question.
My guess is that the father will drop it once he sees that he can't control and bully you - most uninvolved non-custodial parents disappear back into the woodwork when the figure out that getting their way will actually entail things like effort, time and money.
Good luck to you...I would go ahead and accept that job, talk to an attorney or someone who can give you a clear answer on the presumption of custody question so that you know what the worst-case scenario is, and let the chips fall where they may. Even if he tries to file paperwork today, it will take a while for him to complete the forms, have you served, have the return of service delivered to court, and request a hearing, which will be scheduled months from now. So no need to panic today.