https://www.dmv.ca.gov/portal/file/california-driver-handboo...

  If you move, you must notify DMV of your new address within ten days. Submit a change of address online at dmv.ca.gov/addresschange or by mail. It is your responsibility to ensure DMV has your correct mailing address on record.

So, you are required to update your address, and DMV will have your correct address on file, but "you don't have to" update your license.