These are two different procedures. Changing the child's name could be done as part of a custody case; but if you have custody now, you could bring a separate action in court to change the name. The mother would have the opportunity to object - whether the court will listen to her will depend on the best interests of the child. It's the general rule that one parent merely wanting the name change isn't a good enough reason; but if her rights really are being terminated, then that might be enough.
Getting your name on the birth certificate is a separate matter. I don't understand from this set of facts how California's child protective services found you and granted you custody without a finding of paternity having been issued already.
You should consult with an attorney who can discuss your situation in more detail.
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