Your hunch is correct. You should amend your 2022 tax return. The 1099-R you will receive from the 401(k) administrator in Jan 2024 will have a Distribution Code P in Box 7 (Excess contributions plus earnings/excess deferrals (and/or earnings) taxable in 2022.) See page 7 of the 2023 1099-R, linked below.
https://www.irs.gov/pub/irs-pdf/f1099r.pdf
I found this answer to an earlier question. It was on a 2018 excess contribution with a 2019 1099-R with a Distribution Code P
https://ttlc.intuit.com/community/taxes/discussion/1099-r-code-p-what-to-do/00/1442903
A code P on a 2019 1099-R means it was a 2018 excess returned in 2019. The excess must be reported as returned wages on an amended 2018 tax return. Entering the code P in to the 2018 software will ask if it is a 2018 or 2019 1099-R - say 2019 and the box 1 amount will be added to the 2018 1040 line 1 as wages.
