I’d try changing it through the App Store rather than System Settings.
Open the App Store > click your name > Account Settings > Manage Payments > Edit next to the card. Apple says that’s where you can update the billing information for an existing payment method on a Mac.
If the city, state, and ZIP still can’t be changed there, you can add another valid payment method with the correct billing address first, then remove the old one. Having active subscriptions means you need to keep at least one payment method on file; it doesn’t mean you have to keep that particular card.
I wouldn’t cancel the subscriptions just to update the address.