What's confusing me on this machine is the switch. It has four contacts. What seems logical to me is that two contacts would have continuity when the button was released and the other two would have continuity when the button was pushed. That way I would connect the evaporator fan to the contacts that work when the button is pushed and I would connect the hot gas valve to the contacts that work when the button is released. However, when I checked the contacts, I found that there is only continuity when the button is pushed (one pair on each side of the switch). I know that the machine won't cool if the valve is released. I could just skip wiring the valve since it isn't really necessary, but I was trying to make everything work as originally intended. I guess I could also try getting a different switch.
that sounds almost like you're describing an old manual push-to-start florescent lamp switch....
one set of wires is a momentary/normally open ( push to "strike" the lamps filaments) & when you release it, it would "make" the second switch to keep the current flowing to the lamp via the ballast...
But yeah, you don't really need it unless you're in a high humidity environment, these machines were designed to "air-defrost" between cooling cycles ( which is one reason why the Evaporator fans running constantly even in modern coolers.)