Hi Janna,
You can have both Sims 3 and 4 installed on two computers. No need to repurchase either game. They should show up in Origin, ready for download, when you reinstall Origin.
Yes, you can transfer old save games and whatnot as long as you have both your Sims 3 folder and Sims 4 folders saved. Back these folders to a save place like a usb memory stick. So you can move them over to your new computer.
Save these folders -- \Documents\Electronic Arts\The Sims 3
-- \Documents\Electronic Arts\The Sims 4
Note: If you find that you want to play Sims or other Origin games at the same time on both computers. You will need to set one of the computers to offline mode. ["Offline" mode -- In the Origin Client, go to the Origin Menu and select Go Offline.]
In the reference links below it gives information on how to uninstall/reinstall and what files/folders to save for the games so you can use them on the new computer.
Hope this helps,
MrsFlynn
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Reference Links:
Install Sims 4 on Another PC
http://www.carls-sims-4-guide.com/forum/index.php/topic,21899.msg408572.html#msg408572Uninstall/Reinstall (Sims 3)
http://www.carls-sims-4-guide.com/forum/index.php/topic,20732.msg425178.html#msg425178(Sims 3) Saving folders/files to use on a different computer
http://www.carls-sims-4-guide.com/forum/index.php/topic,16414.msg313134.html#msg313134(Sims 4) Saving or Deleting Game Saves (to use on another computer)
http://www.carls-sims-4-guide.com/forum/index.php/topic,21899.msg401675.html#msg401675