It’s recommended your driving licence is prioritised. Updated photo ID is required by some organisations as evidence of your new name.
It’s important the name in your passport is up to date. Full renewal fees are payable regardless of your situation. If changing names from marriage you have 2 options.
You can keep your maiden name at work while taking your spouse’s name on all identification and records. That’s because your married name is considered an ‘assumed name’. This means you can go back to your former name at any time.