Your W-2 should show you and your employer HSA contributions in Box 12 with a Code W. If it does not show employer contributions, and you have documentation of the amount your employer contributed, you can enter that amount in the HSA section in TurboTax, as you suggested. If this applies, you don't need a corrected W-2, though your W-2's should show the employer contribution in Box 12.
It sounds like you did not make personal HSA contributions, since you state the Box 12 amount would be $0 after employer contribution is removed. Does your spouse's Box 12 reflect both employer/employee contributions? Or just employee contributions? If you could clarify this, we'll try to help you further.
You may receive a 1099-SA for the amount that was withdrawn from your HSA by your employer, but if the contribution shows on your 2022 W-2 you should enter it. Your employer may have added it to your Wages in Box 1, so you may not receive a 1099-SA in that case.
Here's more info on HSA Contributions.
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