The only reason to use another washer is to position the valve where it is easiest to get to. My Honda requires the adaptor along with the valve. There should be a gasket between the pan and adapter and another gasket between the adapter and valve.
My vehicle has the plug close to the exhaust and it will be very hot to work in there, especially since it is close quarters to begin with. If the valve lever is straight up and down, I do not think I will be able to get to it without burning my fingers and stout gloves will be too thick to be able to reach in and work the lever. It should be noted that I have the Fumoto but have not needed an oil change since I purchased it so it has not been installed yet.
Emails to Fumoto about this problem resulted in some good tips and correspondence from them. First, the spring lever can work in any position - it does not have to be straight up and down. Also, the washer can be modified and/or replaced with any other kind and material of washer IF you need to position the spring lever differently in order to get to it. The important thing they said was to be sure NOT to over tighten.