I have recently moved from CO to AZ to take in my daughter, who has been living with her father for the past 10 years due to her and his two other children being taken into CPS custody. My court appointed lawyer was for the CPS case but I need a lawyer for custody. Can she also take care of that? If not how do I get a new one?

You can ask him but he appears to be appointed for that specific case, You normally will have to contact legal aid or merely hire your own