me also, i dont wash them until ready to use them, and I just use warm water., and rub with my hands or a soft mushroom /veggie cleaning brush. Its important that the water temp is warmer than the egg, because any bacteria is drawn to warmth.
I use egg cartons and store in fridge they are good for 2 months or so
the older ones are better for hard boiling because its easier to peel them when they are a little older
You really dont want to wash them before storing because any water or cleaner /vinegar/egg wash /anything will wash off the "bloom" which is a thin coating on the shell of the egg that protects it from bacteria entering the egg . Once you wash that off the egg is vulnerable to bacteria entering.