How do I change my billing address w/o changing payment method?

The billing address and card MUST match- this a legal anti fraud financial requirement.

If your move is permanent your bank needs to know YOUR current address !

If you have sorted that out with the bank then you need to decouple the card(s) and update again.

Log into your account in the App Store/iTunes and you should be able to change/verify it there. Or you can click the blue Edit payment methods in the article below. When signed in and the page loads, you will be on the credit card page. After updating that, click the Go Back button at the lower left and you will be on the address page.

Apple ID payment information - change.  

Have you tried logging out of the App Store/iTunes, restarting, and logging back in to see if it will update.