But I knew that wasn't normal, and something could -- nay, should -- be done, stat. So as I reorganized and relined them, I took the drawers out to examine their mechanics. Then I discovered the culprit:
[broken drawer guide]
Most of the guides at the back of each drawer were old and broken, so that they no longer "held on to" the metal rail (seen in the middle of the drawer gap in the photo above). I knew it would be an easy fix and headed over to one of my favorite places: The Home Depot.
[old and new drawer guides]
[new drawer guide, installed]
With minimal assistance from the staff ("Um, where's your kitchen hardware?" "Aisle seven." "Thanks.") I found the replacements and bought enough for all of the drawers. Then, with my trusty screwdriver, I got to work, first replacing the guides for drawers holding knives and silverware (the most dangerous and heaviest drawers, respectively).
[look ma, no hands!]
With the threat of bruised, impaled, or dismembered digits no longer present, my stress level in the kitchen has dropped dramatically. Next step: installing drawer stops.