Part of the problem is that in the current market there is so much competition (multiple applicants for every apartment), I have had clients who have a high credit score and high income get turned down for a rental as well.
Since you are hitting a wall when you mention the voucher, my advice would be to not mention it unless specifically asked and/or when it comes time to submit your application. After all, it is illegal to discriminate based on source of income in Illinois.. so it should be a non issue. The only thing a prospective Landlord needs to know is that your total income is 3.5 - 4 x the total rent.
I would also suggest preparing a letter - have all your information ready to go along with this letter which outlines why you are an excellent tenant (credit score, references, background, etc.) and more information about the mobility program. Try to think of why a Landlord would want to rent to you. For instance, you mention that the program performed a background check on you and it is guaranteed money for them- what else are the benefits? Highlight these points in your letter that will go with your application.
I would also ask your case worker if she has any advice on finding a place. Good luck!
I am surprised that none of the answers here let you know that in Cook County, "Source of Income" is a protected class!
This was done specifically to stop landlords from saying no to section 8!
It is not legal to say no to section 8! If a building has failed the inspection, it would probably be OK to say no, but otherwise they have to treat you as they would any other renter.