I agree with Sondra's statement of not getting CPS involved - that could cause you more problems. I am not sure if the option is available in California, but you might try getting legal guardianship, rather than full legal custody. With an attorney and agreement from the birth mother, you should be able to do that with no problem and no requirements for CPS to get involved at all. That gives you the right to make all legal decisions regarding the child, school, health care, etc. I am not sure of all of the implications, should the baby's mother change her mind sometime in the future, but it is a legal step in the right direction.
Best of luck to you - you do have a lot to cope with.