Ruts are troublesome, that's for sure. I've found the toughest thing about them is resisting my instinctive urge to tense up and wrestle for control.
What works best for me is shifting my weight as far back as possible, like you do when descending. Then let the front end shift around rather than fight it. A relaxed grip and loose arms are key. Yes, the bike will shift side-to-side, sometimes quickly and unexpectedly. But never far enough to go out from under me. The studs help there.
Easy pedaling and high cadence help keep balance too, because it allows me to shift my weight faster in reaction to the bike moving. I granny right down--not because it's tough going, but to keep my cadence high and pedaling effort low.
True, I only go 7-9 mph like this, but with the high cadence, rearward weight, loose upper body, and easy grip on the bars, I can boogie right along with hardly a care.
+1 What he said.
I do the exact same thing; shift my weight rearward and power out of the rut while keeping a light grip on the handlebar. I dropped the pressure in my Schwalbe Marathon Winter tires to 35 psi.