You have no lawsuit against the state or the other parent of the child. You can do many things, however, to protect yourself and your children, which include, maintaining separate bank accounts from your husband; not depositing his money in your bank accounts, filing separate tax returns from your husband, etc. You are not responsible for paying his child support obligations but once you comingle your money and assets they can seek to recover what is there as it is presumptively inseparable from yours at that point. So the best thing to do is maintain totally separate financial lives.
