If she leaves, it would be considered abandonment, but New York is now a no fault state, and it doesn't matter. You should schedule an appointment with a matrimonial lawyer in your community, as there are too many issues to discuss on a message board.
You do not say whether either of you want a divorce. If she moves out with your consent, it is not abandonment for the purpose of divorce grounds. If she does not have your consent, if she moves out and stays out for a year or more, it would constitute abandonment as far as the divorce law is concerned. However, as another attorney has already mentioned, there is now a "no fault" grounds, so it really does not matter if either of you file for divorce. More important is the issue of custody. If your wife moves out and leaves the children with you, this creates a "de facto" custody arrangement in that she is voluntarily leaving the children in your sole custody and care. If she moves out, with or without your consent, you may wish to go to Family Court to seek an order of child custody. The separation papers she wants you to sign should address far more than just her moving out. You really should consult with an experienced family attorney before signing anything.
Mr. Rosen practices matrimonial and family law in Great Neck, NY. His response here is not legal advice and does not create an attorney/client relationship. The response is in the form of legal education and is intended to provide general information about the matter in question. Many times the questioner may leave out details which would make the reply unsuitable. You are strongly advised to confer with an attorney in your own state to acquire more information suited to your particular circumstances.