In order to find out how much of my 2022 estimates state taxes I paid in 2023, would I add together my state income taxes on each of my W-2's?

NO, estimated tax payments are not the taxes that were withheld from your wages.

An estimated tax payment is what you would pay to the IRS or to your state during the tax year using an estimated tax form mailed with your payment or use the IRS or State website to make the estimated payment.

If you did not pay any estimated taxes there is no reason to complete the estimated tax section.

No.   The amounts on your W-2's are not estimated tax paid.  Those amounts are the amounts withheld from your paychecks by your employer, and you only enter those amounts when you enter your W-2's --federal tax withheld is in box 2, state tax withheld is in box 17 and if you had local tax withheld it is in box 19.

If you made some estimated payments by going online to the IRS site or the state web site or by mailing in checks or money orders with a voucher, those are estimated payments.