The state only takes care of kids if a court assumes jurisdiction under WIC 300. You can only voluntarily give ups kid within the first few days of birth (this is known as safe surrender https://advokids.org/legal-tools/safe-surrender/) otherwise courts usually assume jurisdiction due to cases of abuse, neglect, abandonment, etc
Not sure how it works in California but where I live you can sign a voluntary termination of parental rights and if you're in a position to consider it they usually won't argue as far as I know.
