@jenrhaine I can think of a few possibilities. One is that your EA password contains special characters, which Sims 3 doesn't handle correctly. Another is that you didn't remove the data associated with your old password. On a Mac, that would mean clearing all entries with sims3 in the name within Keychain Access, plus clearing Safari's data, plus deleting the [your email].bin file in Documents > Electronic Arts > The Sims 3. Please do all three together if you haven't already.
Another possibility is a problem on the EA end, which you of course couldn't fix yourself. I don't actually know enough about the issue to tell you what you'd need to ask for from customer support, but here's an example of someone else getting it sorted out:
https://answers.ea.com/t5/Technical-Issues-PC/Sims-3-launcher-won-t-let-me-re-download-store-bought-content/m-p/13601174#M264970
If this doesn't help, please let me know, and I'll see if an EA community manager could clarify or at least tell you what to say when you contact support.
Aside from that, Sims 3 can be played without signing in. I never do unless I'm testing something, and other than a few online-only features like Simport and badge tracking, the game is exactly the same without an active in-game login.