"For the long-term excess, I withdrew that by the extended filing deadline in my amended taxes for 2022, so I dealt with it when it was short-term?" - Did you file an extension? Then OK.
"I thought since I then redeposited into my HSA as a contribution for the next year (2023), it wouldn't incur the 20% penalty." - You didn't physically redeposit it into your HSA, did you?
The way it works is that a short-term excess turned into a carryover, is carried over to the next year as a "personal" contribution (line 2 on form 8889). You don't actually put it back into your HSA. However, if you have HDHP coverage in the next year, TurboTax will try to apply this personal contribution to your HSA next year, as IF you had contributed it in the next year.
If this succeeds (because you didn't create a new excess), then the carryover is used up, and all is good.
The 20% penalty is ONLY when you do the worst case solution. See "Fixing Long-term excess" below.
***Notes***
/* Fixing the "short-term excess" */
You must remember to (1) tell TurboTax that you will withdraw the excess in a timely manner, and (2) call your HSA custodian to request the "withdrawal of an excess contribution" (use this phrase) along with the earnings. The custodian should (1) send you a check for the excess and the earnings, and (2) send you a 1099-SA with a distribution code of '2'. NOTE: the 1099-SA will be entered on next year's return, so don't wait for it.
Also NOTE: the excess will be automatically added to Other Income (line 8f of Schedule 1 (1040) or line 13 on Schedule 1 (1040) is reduced) - please do not do anything to report the excess on your 2023 return - TurboTax has already handled it.
/* Fixing the "long-term excess" */
Sometimes, of course, the HSA is empty of dollars because they have been spent already.
In this case, the short-term excess is converted to a long-term excess and is carried over to the next year. Like for the short-term excess, the excess is added to line 8f (see above) or removed from line 13 (see above) AND a 6% penalty is charged on the amount carried over (usually).
However, you can't "cure" the long-term excess by withdrawing it in the same way as the short-term excess.
The cheapest and easiest way is to reduce the HSA contributions for the next year by the amount of the long-term excess carried over. This carryover will print on Line 2 of form 8889 as a "personal" contribution. So if you reduce your regular HSA contributions, the carryover amount will be used up and the carryover will disappear.
Otherwise, you have to make an HSA distribution, and when you enter the 1099-SA, say that it was not for qualified medical expenses. This will add the carryover to Other Income, and add a 20% penalty to your return.
But, at least the carryover will be ended.
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