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Which peice of weatherstrip folds over? Is it the one on the door panel at the bottom or the one that goes around the sides and tip of the glass? Either way it sounds like it needs to be replaced. It is probably ripped or separated from the base of the gasket.
I got some on ebay from carpartswholesale. LMC NPD and JCWhitney have them as well. I like the ones I got. It is far easier to replace them if you remove the door panel. Also, I replaced all 4 inside and out. I think JCW is the cheapest.