GOOD NEWS YOU DONT NEED GRS! for the gender marker change on license. I live in Georgia and went through all of this last year.
ok so like just get an official letter from you doc. My doctor used this letter format
PHYSICIAN LETTERHEAD
I, Physician's Full Name, Physician's medical license or certificate number, Issuing U.S. State/Foreign Country of medical license/certificate, am the physician of Name of Patient, with whom I have a doctor/patient relationship and whom I have treated (or with whom I have a doctor/patient relationship and whose medical history I have reviewed and evaluated).
Name of Patient has had appropriate clinical treatment for gender transition to the new gender (specify new gender male or female).
I declare under penalty of perjury under the laws of the United States that the foregoing is true and correct.
Signature of Physician
Typed Name
Date
and no if you were to move to California, you would just go through their system for the changes. It wouldn't matter if you were born here or not, at least for the license.
If you're talking about the birth certificate. Then yeah you have to return to your birth state to get things sorted I think.