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.
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.
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.