Well, indeed, the culprit was a screw that fell out. In fact, all seven other screws were anywhere from a quarter turn to a full turn loose. I fixed all that and, since I don't have any Locktite handy at home, the current solution is using small squares of Duct Tape to cover the screw bottoms. That's the temporary fix until I find some of those cloth washers or a hot glue gun.