I did as furrydumb suggested. I left the serial code area blank. I then went to My Page (which I could now access,) and, lo and behold, the Sims 3 I had just bought was listed as a registered game.
Unfortunately, my attempts to get stuff through the Exchange throws up an error message saying I have to register a Sims game, so apparently the memo that I'm registered hasn't reached The Exchange yet. It's the weekend... I'll see if things clear up as the work-week starts up.
I used the EA Live Chat help. Had to wait over an hour to talk to customer service, who fixed the issue in under 5 minutes. World Adventures showed up in my game list, the 1000 bonus points were added to my account, and I was able to download the bonus content as well. I highly suggest that you get in touch with customer service. Just be prepared to wait. Have your origin account email and your serial number ready. Good luck!
