The lawyer I hired charges fees in stages, but I can go to the prosecutor's office. What should I do if I want to change lawyers?

According to what you said, you and your former lawyer only reached an authorization agreement during the investigation stage of the public security organ, so his authorization can only be used during the investigation stage.

So when the case enters the prosecution stage of the procuratorate, you can hire a new lawyer and sign an agency agreement with him.