Stiles, ours is plumbed like yours.
But, there is a way you could do a load or two without being connected to sewer, so long as you don't exceed the capacity of your tank. Make sure the dump cap seals tightly, then open the gray tank valve. The water should back up into the gray tank. If you open both gray valves, it should use the capacity of both.
I'm not sure I'd trust that cap, though. I'd get one of those less than $20 slide valves that twist on. CW has them. I have one now and use it for a couple of purposes. With this valve securing the dump outlet in addition to the cap you should be able to run the washer safely with both gray tank valves open. You would have a capacity of 76 gallons plus whatever the dump pipes hold, less whatever was in those tanks to start with.