The drain solenoid burned and overheated, leaving molten plastic on the plunger and jamming it in the out position. Last night, I noticed the same burning smell and removed the kick panel to find the same problem. Since the solenoid was replaced, the machine has probably only received around 7 washes.
There are just two wires that connect to the solenoid: a white/red wire that is shared with other components and a grey wire that connects to the timer. My assumption is that the timer keeps the solenoid powered, causing it to overheat and melt, resulting in the failure.
A faulty timer is the source of your dishwasher's problem. When the drain solenoid was energized, it became caught at the drain point, causing the coils to overheat and melt. This is a fairly typical issue with this GE dishwasher type.