I have lived in my current residence for 25 years and never had this problem before. For the past month or so, there has been a terrible odor that comes up through the shower drain and sink in my back bathroom. This bathroom is back-to-back with my utility room. The odor only comes up into the bathroom while I’m washing clothes.<<
OK Think about this Lint or string is like a wick on an old type of cigarette lighter drawing fluid up by capillary attraction.
In this case lets picture this string in a trap hanging over the crown weir sucking the trap dry PLUS knowing laundry does have a lot of soap scum build up I would strongly suggest you try the following.
Call a Qualified plumber or drain guy and have them water jet your lines.
This water jetting will scrub your internal piping to a like new condition by removing years of accumulation and help remove bacteria growth on the upper piping walls.
While the Jetter is on the job site it would be a great idea to also flush out your vent from the roof to make sure they are in CLEAN working order by removing old scale build up and other debris that fall down these lines TA DUM