What a great storm this is. Being a weather geek, I got busy 2 days before this storm hit. I usually wait until the last minute to harvest my summer carrots. With a spading fork to loosen this up, I put them in the trash, tops, dirt and all and filled it up with water. I let it soak overnight. The next morning, as the storm started dropping snow, I drained the water out, cleaned the bottom and brought it down to the basement. Thus the basement carrot harvest began.
In my experience leaving the dirt on helps preserve the carrots. A thin coat of clay along with lumps is a good thing. I could be wrong but leaving a bit of the top helps also.
I don't have a bona fide root cellar yet. I'm planning on it. For now I'm building a simple wood box and filling it with straw to stick the carrots and other root veggies in. The box will be placed directly under the basement window with the window cracked open and the bedroom door closed. Roots love to be cold, dark and slightly moist. It's best if the moisture is controlled so that it doesn't escape from the box.