The correct procedure would be to amend your 2023 return, showing that you stopped being eligible to make HSA contributions because of Medicare in December 2023.
I have to assume that this amendment will create excess contributions for 2023. You will be unable to withdraw that excess because it had to have been done by April 15, 2024.
Next, since you have filed your 2024 return (right?), you will need to amend your 2024 return as well.
Notes:
1. You don't have to wait to amend your 2024 until you amend your 2023 return - amendments are asynchronous.
2. You will have to wait, though, for the processing on your original 2024 filing to finish before you amend. This is generally when you get your refund (proof that the IRS is finished with the original filing).
3. Because you will likely have a carryover of excess HSA contributions from 2023 to 2024, in the 2024 HSA interview when you are asked if you "overfunded" your HSA in 2023, since this is actually asking if you carried over an HSA excess contribution, you will have to answer "yes", and then manually enter the amount from 2023 (TurboTax would do this automatically, but to make that work, you would have to amend your 2023 return, then delete your 2024 return, then import your 2023 return into 2024, and re-enter 2024 - I assume you don't want to do that).
What is important is the coverage you had on the 1st of the month, so did you have Medicare coverage as of December 1, 2023? If not, you don't need to amend your 2023 return.
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