Legal Question in Family Law in California

You can almost certainly gain an increase in custody based on those facts. If he doesn't fight it, you could get 100%. You should also get a child support adjustment. As for getting "off the birth certificate," that is legally impossible. Being the legal parent not only carries rights, it carries legal responsibilities. You can't waive the rights because they are tied to non-waivable reponsibilities. The law does not allow you to walk away from those responsibilities without either proving you are not actually the father (which must be done within a fairly short time after birth) or having someone else take on the responsibility, i.e., the child is adopted. That can happen either by putting the child up for adoption, or in a divorce situation, by one of the parents remarrying and the new spouse doing a step-parent adoption. So unless you have a husband who wants to adopt your girls, the father remains the father both in terms of rights and in terms of responsibilities - primarily child support.