If you want to determine whether you need to Amend your 2022 and 2023 returns, save a copy of your originally filed returns. Then open TurboTax for those years and Edit your W-2 entry. If your bottom line changes, you should Amend.
If not, just save the corrected W-2's for your records.
Here's How to Access a Prior Year Return if you used TurboTax Online.
As for the university, they normally would pay these taxes for you, so you wouldn't need to pay them to the university. You could check with them just to confirm that.
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