You've rightly identified the cause of the small yield as lack of water. It's water and space that makes the tubers swell, so container-grown spuds are always likely to be second in size to those grown in open ground. However, many, if not most of us do not have acres of garden to use as vegetable patches, and container-grown spuds is the way to go.
As for what to do with them, well the sloppy skins are, as you have said, the original seed potato. The small spuds will grow into new plants if either left to do their thing or lifted and planted next spring, but they will not be certified virus-free and may well disappoint. I wash anything bigger than pea-sized, boil them for 10-15 mins in their skins and then add mayonnaise when cool to give a passable potato salad. Anything pea-sized or smaller gets ditched in the green waste.