The Attorney General Child Support Division is not interested in your immigration status. Nor is the Family Court. You do deserve to know and the child deserves two parents.
This does not establish an attorney/client relationship
A private attorney may be able to move your case along a little faster. Otherwise, it sounds like you are on the right track.
This answer DOES NOT establish an attorney-client relationship. This answer is based on the limited information provided and is not intended to be conclusive advice. There are likely other factors that might influence or change the advice after a more lengthy consultation.
I agree with both attorneys responding to your post. Good luck!
My response herein is an attempt to give you general information and direction and is not intended to constitute an attorney-client relationship as perceived by state law.