Try doing it online instead of visiting the DMV office. They have most of the services online, especially just to change your address.
Go on line or DMV in your area
I'm sorry, I don't know if it can be paid through Western Union. If you are a AAA member, you can phone the DMV to get that answer. I do know that if you want to renew your license though, the change of address must be made first, and then you need to wait at least three business days for it to be processed Before they can renew it reflecting your new address. So, basically it's a two part procedure. If you have your own DMV account, you can check on the status of your address change so that you'll know when to go renew your license.
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