I don't know what's causing the smell, but the moist lint trap and the "takes two cycles" sounds like the exhaust vent is partially blocked.
You can ask the apartment manager to clean it out, or you can take a stab at it yourself with an
electrician's fish tape, a
plumber's snake, or a
dryer-duct cleaning kit. (I picked links from Ace, but you can get them at any hardware or home shop)
Pull the dryer out and detach it from the ductwork. If you're using the fish tape or the drain auger, then you'll want to tie a rag to the end -- washcloth to dishtowel sized at the biggest -- to wipe the lint out of the ductwork. If you've got the duct-cleaning-kit, then use the brush that comes with the kit.
Work your way up and down the ductwork as far in as you can reasonably reach. Chances are good that you'll soon have a huge pile of lint on the floor, and much better airflow through the ductwork.
Hook up the dryer and things should go much better on the next laundry day.
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