I am puzzled by your root rotting and wonder if you had a wet summer? If it is not simply related to soil moisture, then I don’t know the disease or soil parasite causing it.
Dark splotches on washed carrots can happen to most roots after they are kept for too long: I think it is normal, fungal rotting, part of the decay process. It is more sensible for roots to be sold with a little soil on, then they keep much better.