Yes--you enter your 1099R just as it appears. If you want to seek a refund of GA tax that was withheld from your retirmement income you will need to file a non -resident (or part year) return for GA. You do not mention when you moved from GA to IL so it is unclear whether you need to fill as non-resident or part year. The good news is that IL does not tax retirement income so you will not need to withhold IL tax from future distributions.
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Do you mean GA state tax withholding? If so, yes contact your retirement administrator and change your state to your current one. You'll need to file a nonresident GA state tax return to receive a refund of those taxes withheld.
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But also, the prior point about when you moved to IL is important...IF you moved to IL sometime during 2023, then you need to file Part-Year GA,and part-year IL, tax returns...and the portion of your retirement income you received before moving to IL is still potentially taxable income for GA.
____________*Answers are correct to the best of my knowledge when posted, but should not be considered to be legal or official tax advice.*