I cannot find info on this unit but most GE's are the same.

Yes it does sound like a timer problem.

The way that the wash/drain works is that when a drain cycle is required the timer activates the drain solenoid for a couple seconds. It pulls down and the flapper valve is moved to the drain position. The water pressure then keeps it there until the unit is empty and then the spring/s return it to the wash position.

So if the timer contacts stick together due to pitting etc. the solenoid gets power for too long of a time and it overheats and melts the plastic inserts.