I use a Mastercard DEBIT card and 100% online banking.

My bank (Westpac) never sends anything to my old address in Aus.

US vendors insist on a BILLING address as though the Debit card was a Credit card.

How do I get around this problem?

International mail forwarding services (eg BongoUs and MyUS) often provide a 'Personal shopper service', this is not where some middle aged woman recommends you to wear black or vertical stripes to hide your beer gut, rather its a service where you can buy the product via their website and they in turn buy the product for you, have it shipped to their warehouse and forward it on (however be warned they usually add 10$ or 10% for this service, excluding shipping costs!)

Apart from the mark up the only major issue with it, is that shipping also takes a bit longer. Its important to track your order and stay on their case. Another firm who I will not recommend, sent a package bound for Washington, to France.

There are many such service providers, google will reveal all read, review and compare

(Also be wary of the import duty and VAT you might [but not always] incur once it arrives).

" My bank (Westpac) never sends anything to my old address in Aus."

There is your answer, the old address is what the bank has on file and is the address that the account is linked to and is the address that you should use since you are most likely given the option of a "Ship To" address

Billing Address is a security feature to insure that you are who you say you are (in that you know the "billing" address)