First, the code W amount in box 12 is called the "employer contribution" but it actually refers to the amount that the employer PLUS the amount that the employer contributed through payroll deduction. Don't blame TurboTax, it's the IRS' terminology.
Second, TurboTax did not detect an excess HSA contribution because you didn't make one (yet). Therefore, TurboTax did not do the automatic processing to add the excess back to your income.
TurboTax automatically uses the last-month rule, and there is no way to stop that. So what will happen is that next year (assuming you use TurboTax again), TurboTax will notice that you used the last-month rule but did not maintain HDHP coverage. This will lead you into an interview that you haven't seen before which will calculate what your annual HSA contribution limit would have been without the last-month rule, and assess any action based on an excess.
And, yes, there will normally be some sort of amount due because people in this situation usually have excess contributions because the last-month rule gives a higher limit than what they actually merited.
As for the $370 distribution, that's a problem. May I assume that this $370 include the earnings on the "Excess"? Furthermore, since you actually did not have an excess in 2023, this is a distribution for reasons other than qualified medical expenses, which should not only be added to your income but also incur a 20% penalty.
That is, the IRS does not provide a way to withdraw an excess in advance.
The simplest thing to do is this:
1. Contact the HSA custodian and say that you want to return a Mistaken Distribution. This is how you will return the $370 to your HSA. Note that the form for doing this is often online at the custodian's website.
2. Next year, TurboTax will discover that you did not maintain HDHP coverage and will ask you a lot of questions about your 2023 contributions and income and whatnot. Then whatever you owe will be added to your tax due...and that's it.
No special calculations on your part or special manipulations of your HSA.
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