You should not claim credit for taxes paid to IL (on IL income) on your GA tax return, unless you are also paying tax to GA on the same IL income.
You can only claim a credit for taxes paid to another state when both states are taxing the same income.
If you were a part-year resident of each state, you should only pay tax to IL on income earned while a resident of Il, and pay tax to GA on income earned wile a resident of GA.