Mailed state return (IL) without signature. Should I send a second one with sign?

No, do not send another IL return.

If they notice that the return is unsigned, they will send it back to you to sign.

Otherwise, there's no problem.

Thanks @BillM223. Would I be charged a late filing penalty if they notice that my return was initially sent without a signature and then returned to me for signing? 

I don't practice in Illinois so I can't say, but I doubt it. They know when you originally mailed your return.

Besides, a second filing would just confuse them.