Put them in a wood box. Lots of air holes so they can breathe. No light. This will help regulate the humidity and prevent greening. They will last substantially longer.
Also there is no rule saying you can cut each sprouted part into a three cubic inch piece and plant them in your yard. After you have big broad leaves on foot long stocks pile up about four inches of soil out to where the leaf tips are. Once the stalks collapsed and turn brown carefully dig through the soil. You have discovered the unlimited potato glitch.