Great question — answering as the UphillTech team:
This is probably not what you want to hear on any front, but it is best to store your eggs in their shell. Their original carton provides an ideal container. If you do want to store them peeled, the standard way to do it is refrigerated in a bowl of water (changed daily). Of course, you will then have to dry the egg before eating it the wetness bothers you. This method is also used with poached eggs. Lastly, you can do what you are already doing, and store them with some towels, but they will probably express some water. The eggs should be good for about one week, unpeeled. The Egg Board recommends they be eaten the same day they are peeled, but other sources indicate up to about 5 days....
Full answer by SAJ14SAJ: https://cooking.stackexchange.com/a/35921 (CC BY-SA 3.0).
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