I'm definitely not an expert, but I do have personal experience in this area.
My suggestion is to start by shaking the plants to remove excess dirt and insects. If possible, shake them out over an area where plants are growing so the insects will have a new home and the fertile dirt can go to good use.
While shaking the greens out, gently separate the leaves a little (without breaking them from the stalk) to let trapped dirt fall away. Keep the plants as intact as possible until you are ready to eat them.
Hold off on washing anything until you are ready to eat.
When you are ready to eat, fill a small bowl with water. Then separate out what you are going to eat for that meal and rinse them in that water. You can use your fingers or a gentle brush to push away any stubborn dirt. You'll be able to use the same small bowl to rinse all the greens for your meal.
Experiment to find the right shape of bowl for the job. I prefer something similar to the shape people often use for eating oatmeal or cereal.
If water is extremely scarce, you can even have a second small bowl and filter the water from the first bowl to the second bowl to remove some of the sediment. Especially with refrigeration, you may be able to use that second bowl the next day.
When the water is too dirty to use for rinsing what you eat, you can use the water for other purposes, such as watering your crops or as part of the process in making bricks or pottery.
One of the important aspects of this technique is to start with good clean organic soil that is free of contaminants. For this reason, you want to be sure that carnivores (including dogs and coyotes) are not defecating on the soil anywhere near your crops. If the natural water table is being used as a water source for crops (unlikely in your case), you'll want to be sure it is safe to use for that purpose.
If you're unsure about health/safety stuff, I recommend getting help from experts, especially those familiar with the water, soil, and crops in your specific region.
