It is hard to tell where the wires are going to from your pictures and this is where a shop manual comes in handy. I can see that your solenoid is three prong which means there is a holding circuit after the start circuit. You should also be checking that the relay/time delay control module is working properly and sending signal when it should and that the hold circuit is energized as well. Perhaps someone here knows your particular machine better than I do and can chime in.
Another thing you may want to check is if the lift pump is actually creating pressure.
Here is a video that you may find useful to test the solenoid.
Hope this helps!!