I store mine in 5 gallon buckets with damp wood shavings and the lid snapped on the bucket. I do a layer of shavings, a layer of carrots (on their side, tops trimmed fairly short, not touching one another), a layer of shavings, carrots, shavings, etc. It sits in my root cellar which is just above freezing (if it ever gets cold this year). I don't use sawdust, I think the particles would be too small. My wood shavings come from a woodworking planer so they are pretty small, definitely not as big as wood chips from a brush clearing operation. As for dampness, I try to get them less damp than if you got them totally wet and squeezed all the water you could from them. So they are probably just wet enough that you wouldn't expect them to soak up a water spill. Hopefully that makes sense. Damp but not fully saturated.
Two winters ago we used damp sand which worked but made the buckets really heavy and then we were washing sand down the kitchen sink drain all the time.
This method works for carrots, beets and parsnips and I'd assume other root crops.
I think I would avoid anything that is "partially decomposed" just in case the microbial life on it can overpower the carrots. If you look around for a woodworker with power tools you should be able to get "planer shavings" pretty easily.
They do get sweeter after a solid frost. I think I would leave them but I don't know. A carrot's goal in life is to make it through the winter to produce seed the second year. So if they're rotting, it's probably due to the soil being too wet for too long or maybe a pest that damaged the carrot root. I'm not sure.
Compost is filled with rot inducing microbes. I wouldn't use it to try to store vegetables....
In my culture, we traditionally use sand for storing root vegetables. I often bury carrots about 18" deep in the garden in clay. Some of them survive, some rot.
Carrots will store well in the ground until it gets cold enough to freeze them. Some years, they might survive all winter in the ground, other years they freeze by November. I suppose depending on how much snow is covering them when it gets super cold. Some years we throw straw, leaves, or dirt over the row to try to hold them till spring.
In my family they are stored in peat moss in a cold room. We use those big square rubbermaid containers, with a lid. A layer of peat moss, a layer of carrots with the tops cut off. Layer of peat moss, etc.
The carrots are stored in the cold room along with cardboard boxes of potatoes. The cold room is kept around 5-10 C by insulating a room in the basement from the rest of the house and ventilating it outside.