I've found that there are a lot of strong opinions about whether or not to wash eggs. Here's what I do. I buy a concentrated natural egg wash at the farm store. I have a spray bottle that I mix 2 capfuls of the egg wash concentrate to 1 quart water. I keep the spray bottle by the kitchen sink. When I bring the basket of eggs in, I spray them with the mixture, and then rinse them off with very warm water. I pat them dry with a paper towel and immediately put them in the refrigerator. If there is poo on the eggs (not very often), I use an antibacterial dish detergent, too. I never submerge the eggs in water, though - and I always use very warm water. This is what works for me. The eggs stay fresh for weeks, and I always know that every egg in the refrigerator has been washed. It just seems like extra work to wash/rinse them as you use them. I'd rather go ahead and wash the day's worth of eggs at one time and be done with it. I read that it takes a refrigerated egg a week to reach the level of deterioration that an unrefrigerated egg reaches in one day. So it makes sense to me to keep them refrigerated.
But - everybody has to decide what works for them. There's no right or wrong.