Why does having the furnace on produce a sewer smell ?

So in a way I think you are correct. You don't have a window cracked so stinky air slips up past a faulty air admittance valve. :B

Hopefully you can find someone at the park with long lanky arms. I'd do it for you if you were anywhere near me. Oh ans here's a picture you can show to the guy at Home Depot to help locate a replacement.
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