I had a similar problem on my 68 and eventually just replaced both handles as part of my restoration, however before that time I fixed the problem by removing the handle and building up the area behind the tang that is attached to the moveable flap that your fingers touch when opening the door. Hard to explain but when the handle is out of the vehicle, it will be very visible and you will see that just adding a bit of filler with JB Weld or even a thin metal shim BEHIND the tang that hangs down, will adjust the resting position of the chrome flap that is now sticking up in your picture when the handle is reinstalled into the door. Sounds more complicated than it is since on a 68 the handle is not attached to the door linkage and is only held in place by one attaching nut. It changed in 69.
Another way would be to slide a small section of tight fitting rubber tubing over the tang and that will also act as a "shim" and thereby adjust the chrome piece that is protruding in your picture.
Use adhesive so the rubber tube or hose does not slide off and change your adjustment or fall down inside the door.
Kurt
