@MeganFord1213 The way to find codes in Windows is to look in the registry; here's a guide:
https://sims3.crinrict.com/en/2013/09/how-to-find-your-serial-number.html
But depending on when you bought the game, you might not have received codes. In that case, you'd still need to contact EA support. If they weren't here today, they should be back tomorrow, but I can check on that for you. It would also help to set your location to U.S., if you haven't already.
The workaround for packs that won't download doesn't remove Origin from the process (and besides, you need to install TS3). What you do is run the Origin reset tool for macs, which removes its cache files and saved preferences. Then you sign in and uncheck the box to automatically keep Origin and your games up to date. Then, you go into Application Settings and re-enable automatic game updates, and quit. When you sign in again, Origin will automatically offer you the downloads of each pack you haven't installed.
Of course, to install those packs in mac OS, you still need your codes. But whether you can find them in your old computer's registry or not, you still own them, and EA support should provide them when asked. If you have any trouble with this, let me know, and I'll try to get someone more official to respond.