I cannot attest to the rules in Portland but here in the Lewiston-Auburn area there are some issues with GA.
For instance, GA does not allow for a security deposit and does do not pay Fair Market Value. The law states that a landlord MUST accept Section 8 and city assistance vouchers but they do not have to lower their rental amounts.
It is illegal, as a landlord or property manager, to accept an "under the table" security deposit or to require (and accept) additional money above and beyond the payment made by GA (or Section 8).
I suspect that this is what you are facing. I know this doesn't help except to say that you may be looking at rental property that is of higher value then your GA allows for.