I had this ice-partially-melting-in-the-ice-container in my Kenmore/Frigidaire side-by-side, in-the-door ice dispenser.
I used a flashlight and looked up at the outside of the flap-door, and sure enough it was gaping open by about 3/8 of an inch or so. Pushing against it would close it, but it would spring right back open, so something was wrong with the whole flap-door mechanism.
Fiddling round a bit, I found that the clear plastic ice shoot had become dislodged, so the shoot wan't being held in place like it should by having its top wedged behind the back of the top of the bezel that surrounds the whole exterior dispenser unit. Turns out the wedged shoot is what holds the entire flap-door mechanism in place and that was why the flap-door wasn't shutting all the way. As soon as I snapped the clear plastic ice shoot back into place, the flap-door worked perfectly!