After I filed my federal return this year, I realized I forgot to list about $350 in federal income taxes that was withheld by my bank on a savings account. So I am owed another $350 by the IRS. My question is whether I should file an amended return or maybe just report the extra $350 tax I paid on next year's tax return.
Wait until the IRS finishes processing your tax return. There's a good chance that they will notice the omission and correct it without your having to do anything. If they don't, then you will have to file an amended return.
You cannot report tax that was withheld in 2020 on your 2021 tax return.