He can petition for custody, but he will have to prove it will be in the child's best interest to reside with him. I suggest you retain an attorney to file a custody action on your behalf, just to make sure you get an official order outlining a custody arrangement.
If you never signed divorce papers, filed them with the court, and received a signed JOD from a judge, then you are still married. You can not get remarried until you divorce your wife first. I urge you to speak with an Erie County divorce lawyer before moving.
There is probably a good reason why your lawyer is pushing for trial. You need to sit down with your attorney to discuss your concerns, and get a better understanding of why he or she wants this to play out in court.
If you still do not feel comfortable with going to trial, you can always hire a different attorney.
You husband's attorney should serve you a Notice for Discovery and Inspection, and a Demand for Interrogatories. You need to respond with whatever documents are requested (bank statements, tax returns, etc.), as well as answer any questions on the Demand for Interrogatories. As stated previously, these can be general items or questions, but also more case specific depending on the nature of the issues at hand.
It may be in your best interest to consult with an attorney if you do not...
Before you sign anything you should consult with an experienced divorce lawyer. If you sign the papers and your husband obtains a divorce any money or property in his name will be his. Any debt in your name will be yours. If you waive maintenance(alimony in NY) you will never be able to go back to court and ask for it. If you have children with your husband, any custody or support obligations will be affected by the papers he may try to have your sign.
Your situation is very common....