From what you are saying I would suspect the water level pressure switch is at fault .
Here is a link to the part .
Then a second connection as you call it makes and advances the timer /control to start agitating.
So it fills and then does nothing elses ,it is the pressure switch .
All GE HydroWave washers have a shaft and tube as opposed to the old transmission. How this works is there is a sifter that shifts the washer from wash to spin mode. This is done with the sifter coil. The sifter coil is basically a solenoid that pulls up a set of gears that lock into another set of gears for wash mode and for spin mode it simply disengages. In other words, with the gears disengaged the spin basket turns with the agitator and with the gears engaged the agitator spins independently from the spin basket.
When the coil fails it can no longer engage the gears, therefore the gears stay disengaged all the time. This will cause the spin basket to turn with the agitator even during the wash cycle. This will make the washer become noisy during wash and it will not wash as it is designed to. Not to mention this is very rough on the machine in general because it is not designed to handle a full tub of water turning back and forth continually. Also this will cause motor to flag the four-flash error code, which stands for a failed shifter.
If your washer has the symptoms I described above more than likely the sifter has failed but there is a very slim chance the motor is causing this problem by not powering up the coil for wash mode. Check the sifter coil and if it is bad replace it and if not replace the motor. Follow these steps to check the sifter coil.