Wash your dishes in the bathroom sink of the local park.
If you eat a water-based diet, washing dishes is very easy. Pour enough water in a pot to cover the bottom, scrub it with a brush while swirling the water around, then toss out the window. Repeat if necessary. No dirty hands, no soap or hot water necessary. Baking soda helps with light grease and crusted-on food, but I've never had to use it.
If you use grease in cooking, first wipe out your pot with a paper towel, then add baking soda and a little water to your cooking pot and bring it to 180 degrees F or so on your stove. The baking soda will undergo a chemical reaction and become sodium carbonate, which is a very powerful degreaser. Scrub and dump as before. Rinse once with a small amount of clean water. Unlike dish soap, sodium carbonate is non-toxic in small quantities and does not make a mess. Save the greasy paper towels for starting campfires.
On the road, many of us have very limited resources and have to leave sticks n bricks habits behind. The complete table setup with silverware and tablecloths and saucers and glasses and centerpieces is reserved for stationary housing. When cooking on the road, there is nothing wrong with eating right out of the cookware using the cooking utensil, drinking out of a canteen, and washing dishes in a creek or bathroom (as long as we don't throw the dirty water back in the creek or clog the sink).
Just a note on reusing graywater for plants; the food particles in dishwater can produce a foul odor when decaying, so it is not ideal for indoor plants.