Congrats, you've just learned an important lesson about winter driving.
Basically there's two ways to drive across deep snow. You can plow through it or crawl slowly on top of it.
Even a lifted Liberty has limited ground clearance and if there's ice beneath the snow it compounds the problem. Aggressive tires, even studs won't help. I haven't tried driving my Liberty on top of the snow because it's pretty heavy and the tires aren't that wide. On the other hand I've driven my YJ on many snow runs on top of the snow. The YJ is pretty light and the tires are 12.5" wide and with the tires aired down to 6 lbs they act like snowshoes.
Of course sometimes even my YJ gets stuck

Thank god for the Warn
The next time you try to drive through deep snow I suggest you do it slowly to ride on top of it or rock yourself forwards & backwards. It may take a foot at a time but you should make it.