Wood ash or sand grit works great on hard packed snow or slippery surfaces. Tuga BURIED his wheels and if you northern boys want to think of an analogy, imagine your bus body hung up on wet heavy snow while the wheels spin down through that stuff. You can provide traction till the bovines arrive but what Tuga needed, and got by waiting was a hard surface capable of supporting the coach drive tires.
Tuga, all the differential does is allows the wheels to travel at different speeds so when going around turns you are not scrubbing one set of wheels. The downside to a differential is that it will supply power to the set of tires that has the least traction. Newell did a great job providing the differential lock so you could get traction to both, equally and simultaneously.
In some trucks you can lock the wheels side to side (as you did) and also you can lock both sets of drive axles.