I think you'd better just use those damaged potatoes right away. If rot starts in that box of spuds . . . well, you know the old saying, "One bad
pomme de terre spoils the whole barrel

!"
I don't have too much trouble with weeds in the potato patch. One thing, I seem to be keeping a compost pile just for the potatoes in recent years. Right about the time that the weeds in the spuds could really get out of hand -- it is time to hill the potato plants!
Instead of gathering dirt around them, I use the compost. I start off with some organic fertilizer then, after layering on the compost, the weeds are smothered, any potatoes near the surface are covered, and the plants are fertilized. The insulating compost probably also keeps the soil cool and helps conserve moisture.
To keep damage to an absolute minimum, and since my potatoes are planted in a bed, I start digging at one end of the bed and dig the entire thing out to a depth of 8 to 10 inches. The tubers are all in about the top 6 inches of soil so all the shovel-work is below their grade. They just fall into the trench as I go along. It takes me weeks to dig out a few hundred square feet but there are different varieties that mature at different times so, I don't need to be in any hurry.
I wash the harvested potatoes and allow them to dry in the shade before carrying them downstairs to storage in the basement.
Steve