The prompt that comes up to initiate Apple Pay has a section for the shipping information. If you click that, you can create multiple locations and addresses for each location. You can even create custom emails in the same way under the email section.

Is that what you were doing and were unable to do it or was there some kind of technical issue?

I just hit "Buy" and it was bought -- I didn't get any prompt that I noticed except for "approve this on your watch"

There's a prompt after you hit buy where it requires you to either double-tap the side button (on iOS) or TouchID (on MacOS and some iOS devices). On that prompt, it shows all the transaction information including shipping info, the card Apple Pay will run the transaction on, and the contact information for the transaction.

You must have been speeding through and missed it.

I'm supposed to enter the address on my watch?

You didn't pay attention.

I have shipped to other addresses than my main address on file with Apple.

blame the user!