If there were not any state taxes withheld and reported in box 11 of your Form 1099-G, then you do not need to complete boxes 10a and 10b. Leave all three boxes blank, even the drop-down box showing your state.
If you are using TurboTax Online, there is a place to enter local income tax withholdings and locality name. Only enter this information in both boxes if local taxes were withheld. If your form only shows a locality, but no taxes withheld, then leave both boxes empty. (If you are using a CD/downloaded version of TurboTax, there is no place to enter any locality information. That information would need to be handled separately when completing the local tax return if applicable.)
[edited 1/25/2021 | 2:12 pm PST]
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