You do not need a corrected W2.
For the Georgia return, TurboTax will calculate how much you owe Georgia and apply the $2000 (so you may have a balance due or refund from Georgia). Then when you do the Ohio return, TurboTax will calculate the tax due, subtract the Georgia amount and apply the Ohio withholding.
For the Georgia income, the money you made is what you got paid for working in Georgia (so the $9728.15). @amazingtax2021
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