That's all I do... but I do that with all of my eggs, not just the visibly dirty ones. Gentle washing by hand with water that is just about the same temp of the egg, maybe a tad warmer. You can buy "egg soap" but I have no use for it. The only reason I wash them at all is because I am not the best in the world at cracking eggs and I don't want a broken shell with chicken doodie on it falling into whatever it is I'm making
eta - to clairify - I wash all of my eggs, but that's right before I use them. I don't wash them until then.
Just use water that is hotter than the egg. This causes the pores in the eggshell to contract and pushes any dirt to the surface. Then dry thoroughly.
Washing eggs washes off the bloom - a coating, if you will, that the hen puts on her eggs. The bloom protects the egg from contamination and they stay fresh longer, so use any eggs that you have to wash first.
Yes, the temp. does matter. Using cold water will cause the pores in the eggshell to dilate meaning any dirt on the surface can be pulled into the interior of the egg - the part you are eating.
I don't sell eggs to my customers that are more than a month old, although I never have eggs around that long.
For my own family's consumption I wouldn't have a problem with eating an egg that's a month and a half old, so long as I haven't washed off the bloom.
It's possible to buy eggs in the grocery store that are at least 6 weeks old.
ETA: The contracting/dilating thing may be reversed, I'm very tired, but the concept is the same. Use water hotter than the egg itself.
Do not wash eggs in cold water. The contents of the egg will contract and pull bacteria right into the shell. Water helps bacteria get into the shell whether cold or hot. Using hot water just lessens the risk because the contents of the egg expand pushing moisture out and not letting as much get in. Eggs also have a coating on the outside that prevents bacteria from getting inside and that washes off easy. That's why many don't wash their eggs and personally any egg I have that gets wet before I'm ready to use it I feed to the animals. If eggs are really dirty I just don't use them. If I were going to keep them I would wipe the outside with a dry cloth to get the worst off, put them in the fridge, and then wash them in hot water right before I was going to use them.
An experiment was done that showed not washed farm fresh eggs were still good after 6months in the fridge. Dirty eggs washed in cold water I would use within a couple days but then again preferably I wouldn't use them at all.