Yes, you need to amend your state returns. You will need to make the changes on the federal return, but you won't have to file the federal amendment.
You need to add a nonresident IL return. How do I file a nonresident return. The 1099-G should be from IL - that income and the tax withheld is IL sourced and should be reported on the Illinois nonresident return. When you go through the nonresident return, TurboTax will ask you which income belongs to IL. Your resident state will tax all of your income, but Wisconsin will give you a credit for tax paid to IL.
How to Amend
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