State tax filing

No. Enter the W-2 on your FEDERAL section of TurboTax (only) as it was reported to you.  Don't CHANGE anything on your W-2.

AFTER you complete the FEDERAL return, start the Michigan non-resident return.

TurboTax ASKS you what you EARNED in Michigan, answer 0.

See screen-shot

TurboTax will see the Michigan tax that was withheld and reported on your W-2. 

TurboTax should then REFUND that amount back to you. The local tax might be adjusted, but it won't be completely refunded.

Finally do the Illinois return. 

Allocate all your income on the Illinois return.