I'm having a similar problem, but I suspect it may be due to missing parts. My truck is a 1990 AM General with a soft top. The screw heads are located behind the rubber weatherstripping, and it gets "peeled out" when rubbing against it when getting in the truck or opening and closing the door.
Our volunteer fire department received a 1984 AM General M923A0 recently and I was asked to help repair it. It has a hardtop. Upon looking at the truck, I noticed it has small thin strips of metal with screw thru the metal holding the window weatherstipping down. Looks like factory parts and fits very well. Not one hint of trouble compared to my truck.
Can someone help me figure out where to get these metal strips and what the proper name of the part would be? I would appreciate it if others will chime in and state if their personal trucks have these or not.
Pics of my truck without the metal strips. I will post pics of the fire department truck later.