turnip/rutabaga trouble

Sorry to hear of these problems!  I don't grow rutabaga, so not any help of that, but haven't had problems with storing turnips.  With turnips, leaving the greens on the root will cause it to rot a lot faster. I cut the greens off at the base of leaves, as close to the turnip as possible.  Store the unwashed turnips in damp sand or sawdust (not wet) - fill a container part way damp sand, push the turnips into the sand with the tops down, making sure the turnips do not touch each other, then cover with more damp sand. You can have multiple layers of turnips in the container as long as each layer is completely covered with damp sand and the turnips do not touch each other - it is best to have about 2" between the turnips in order to ensure there is plenty of damp sand around them. They will keep for about 6 months this way.

Turnips can be stored in the fridge crisper drawer for about 2 weeks if the tops and roots are removed, and they are washed with cold water to remove all dirt. If they are kept in the fridge for more than a couple weeks, they will get soft and mushy.