I had this and fixed it through several steps.
If you go to settings on the phone and then into Wallet & Apple Pay you will find transaction defaults. The old address was here, however changing it to my new address did not fix the problem, also if you add it here before fixing any other issues it just keeps coming back.
I also had to update My Card in Contacts, this had two address in it, the old and the new. This seems to be the key step for me. Removing the old address now gave me only one option in the Wallet & Apple Pay default settings.
At this stage I also noticed that if you go into the wallet and click the i on the credit card you are using you can change the billing address, again both addresses were here. I again deleted the old address. The first time it came back! but doing it again deleted it. I am guessing removing the card and adding it again may also fix the problem.
Then I for good measure I logged out and logged in using iTunes & App Stores in Settings and restarted the Apple Store App that I was using.
Now my delivery options show the right zip/postcode and on check out with Apple Pay the address is correct.