Okay, I sell quail eggs for eating. Do NOT use the same cloth to wipe all the eggs or use water that is colder than the eggs, as these can and usually do introduce bacteria.
Best practices follow:
Refrigerate eggs as they are collected, (do not wash before).
Set out your clean fresh cartons and a bowl of warm water and plenty of paper napkins.
Pull cold (40F) eggs out of the refrigerator, and use paper napkins to wipe each eggs, if necessary dampen napkin in the warm water. ( DO NOT DOUBLE DIP). Use a new napkin (or two) for each carton at the very least.
Place freshly cartons eggs back in refrigeration as soon as they are filled.