Weird

I would remove the door's inner panel and check the wires, especially where they bend to go under the unit.

Although your unit is not listed in the recall also check that the rinse aid has not been leaking down onto the wires.

Only way to tell if the option switches can do this is by looking at the wiring diagram. Otherwise you are limited to a visual inspection.

Do you have the wiring diagram?
If not check under the unit to see if you can find it.

I have a similar model and the wiring diagram shows that a set of timer contacts choose either the drain solenoid or the fill valve but there is no way for both to be on at the same time.

But yours may be different and finding GE wiring diagrams on the internet is next to impossible so it makes it difficult to diagnose problems especially weird things like you have.

You would have to catch it when it does a fill/drain at the same time to tell if the wash/drain flapper valve is just sticking or is the solenoid being actuated.
Cannot say for sure but on most GE units like this the wash/drain solenoid is just activated by the timer for 20 seconds or so. The water pressure then holds it in the drain position till the unit is empty. The spring then pulls the flapper back to the wash position. The solenoids will overheat and melt if they are powered for the full drain cycle.

I played with the flapper through all cycles, and it moves freely. Does not bind.
If you are in a wash or a drain cycle the solenoid plunger should bind because the force of the water on the flapper should hold the flapper either open or closed depending on the cycle. Almost sounds like the flapper is not attached to it's shaft.

then the water shuts off the the solenoid should close before the main wash cycle begins, then the water comes on again and the water drains out.
Am a bit confused here.
When filling the solenoid should be at home position (plunger up). It should stay there throughout the wash. When it reaches the drain part of the cycle. Then the solenoid should activate (plunger down). Then it drains till empty, solenoid returns to home position (plunger up) and you are ready for the next water fill.

If the water drains out when the solenoid plunger is up then you probably have a flapper valve problem.

Here are your parts
Model GSD3735D01WW

The water started to fill and the plunger was clearly up.
This is OK

After the initial was cycle was done, the plunger went down, the tub drained, and the plunger stayed down. The water came on and continued to drain. As the water continued, you could here the dail click to turn off the water. The plunger remained in the down position.
Here is where it messed up, once the unit had drained the plunger should have moved up.

After the dial reached the rinse cycle, the plunger switched to the up position, filled the tub and went through the rinse cycle. At the end of the rinse cycle, the plunger swithed down and the water drained.
This is also OK

You have to determine if it is an electrical (timer) or a mechanical fault (sticking plunger/flapper)

Is solenoid getting power when it should not have?
After the initial wash cycle it should get power for about 20 seconds, plunger pulled down, unit drains, plunger held down by the water pressure, unit emptied, plunger moves up.
It should not get power again till the end of the second wash cycle.

Best way to monitor it is get a meter, attach it to the solenoid with everything in place. Often you can just push the meter leads into the back of the solenoid connector. Put the meter on a 120 volt scale and let it run through a cycle.
If it does get power when it should not then probably the timer is shot.

If you do not have a meter you may want to purchase one. You can get a decent digital multimeter for under $20.00, You do not need fancy but it is nice if the leads are a couple feet long. If it saves buying one unnecessary part it has paid for itself and you have a useful tool.

You could also run a cycle while pushing on the plunger's linkage to hold it in the up position. When the plunger gets power it pulls down good and hard so you will know it is powered. If by doing this it returns to the home position where it now stays in the drain position then you know it is a mechanical problem.

Could be the return spring on the plunger is not strong enough.
Could be it is not draining properly. A common cause of this is the Piston & nut Item 564 in the "Body parts" section. This is under the cover at the inside back of the machine. Once you remove the cover it unscrews (reverse thread, I think). Normally though you would see that the unit is not draining properly. This unit does leave just a little water in the bottom with the sump full. The pump does not like to run dry, in fact it will burn up the seals if run while dry.
Could be a partially plugged drain line.

Hope this helps and makes sense.