You should wash your eggs in WARM water. I just keep a scrubby by the sink specifically for that purpose. I let them air dry beside the sink and then pack in egg cartons.
For me ~~ I keep my eggs unrefrigerated if they are unwashed (They have a natural bloom that helps protect them). If they need washing, they need to go into the refrigerator shortly thereafter. I can keep unwashed/unrefrigerated eggs in my back room for several weeks during most of the year...