I have 2 kinds of wetbags (well, we're currently potty trained) - the Wet Happened bags sound like the ones you're describing where the lining is attached up by the zipper. I don't think this is technically considered PUL, but just a water proof material. Mine is very thick and durable, so I would just toss it in the drier inside out (or plastic liner side out) - though, as you described - it usually was separate from the cloth part and twice it's normal size.
The other kind I have are pail liners with actual PUL (similar to what's in diapers like Bum Genius) and has an elastic top. These I've noticed get worn out in the drier so I would just hang them dry. It's faster if you turn them inside out (PUL side out)...
And on that note - In my experience, your diapers will last a LOT longer if you line dry the covers (assuming you're using the PUL lined all in one types...). A friend of mine dried hers all the time and they only lasted about a year or two before the PUL started to fail.
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