Q. Should I just ignore the Res W2?
A. Yes, probably.
If you worked in Cincity, you cannot get any of the box 19 amount refunded. On the hand, if you itemize on the federal return, the box 19 amount is deductible.
You say you are an IL resident. That type of W-2 split would indicate that you were a resident of the city for part of the year and continue to work in the city, as a non resident for another part of the year. You may need to ask your employer for an explanation.
For help, in this forum, we need a better explanation of your situation.