There has been a lot of trials done on storing wet or modified wet distillers. It is a very dense feed, and for bagging you would need to mix a ground forage in with it to keep the bags from splitting. 
In a covered pile, you will get more longevity than uncovered but it will have to be airtight to prevent spoilage. It will obviously keep better over the winter than in the summer. I would avoid using ag lime to cover plastic unless you are careful not to get it into the ration; there are reasons we have feed grade lime, which include avoiding potential toxic contamination (which can reduce performance and kill cattle) and food safety issues (which as a food producer would put you at risk legally).
With 60 head, a semi load of wet is going to last you about 3 months. In my experience, dried distillers grains are often priced competitive enough to not mess with the wet on this number of head.