This happened to me...

[TheMightyBowHunter]

Once upon a time, I was going bird hunting with a shotgun. I decided since there  was a fresh 18" snowfall, I would try grouse hunting on cross country skiis. So I got my gear on, and grabbed the shotgun, but wondered what to do with poles. So i left them stuck in the snow. After struggling for a few hundred yards, due to being tippy, I had shoved the barrel in the snow something like 19 times to keep my balance. At that point I realized shooting would be a bad idea, so I turned around and headed back to car. Anyone ever do this successfully with a sling of some sort?

[Logan7]

Not me.. but I think what you're describing is almost the sport of biathlon. That's cross country skiing and target shooting. They wear back slings.

https://www.olympic.org/biathlon

http://soldiersystems.net/2015/07/05/vss-services-tactical-backpackbiathlon-rifle-sling/

[TheMightyBowHunter]

Biathaloners carry a rifle on the back. When a grouse takes off, the shotgun has to be readily available in a second. maybe snow shoes are a safer choice!

[JimmyJamesKY]

Sliding across the snow balanced on two slick boards with a loaded shotgun sounds hazardous! 

[thresher_593]

The Biathlon uses different shooting positions at different stations. Some are standing, some are prone. The rifles are all .22 Long Rifle so there is no recoil to worry about. In international competition the stations have a rubber or similar surface. In the standing shooting form, the rifle is perpendicular to the skies so, even if there were recoil from the rifle, the  chances of falling aren't too high.

Little different story with a shotgun I believe.