How do I go about getting a new lawyer for my appeal?

You or your family would have to contact the attorney and make an agreement. You must realize that the private attorney has to be paid by you or your family. Even if your family retains a private attorney it may still be possible for your attorney to receive a free transcript of the proceedings. I have been successful in making such a motion even when I was retained by the defendant's family for the appeal. As long as the defendant is indigent the court would likely provide a free transcript.