It's not unusual for states (and sometimes IRS) to be a couple of days late direct debiting your account. But 12 days is excessive.
TurboTax would have no info about the status of direct debits. The PDF of your tax return would contain a sheet, "Electronic Filing Instructions" that would contain info about the method you chose to make payment of your tax liability.
I'd for sure contact NY State Department of Revenue to see if they can offer any guidance. The IRS won't be available for telephone info. They are so short-staffed right now.
The only way to "cover" yourself, of course, is to make payment again. This would stop any penalty/interest accruing. But since you've said the amount is quite large, this may not be an option for you.
Here's info for contacting NY DOR:
Contact us (ny.gov)
And if you decide to call the IRS, use the info I'm sharing below.
How do I contact the Internal Revenue Service?
*** I am NOT a tax expert. I am a seasoned TurboTax user, and volunteer to provide assistance to TT users. Nothing I post is to be considered TAX ADVICE; I bear no legal liability for responses.***
