Klink : I did the nearly the same thing .........
As I said, almost the same. Except, no ice under the snow, but the snow was actually deeper.
My driveway is over 700' long and in the middle of ski country. Down near the end of it I have a large lateral turn around area (none of this area of the drive is paved... it's gravel and the turn around leads to the rest of my land which would be considered "off road")). I use a mid-size diesel 4-wd tractor for snow removal. In the turn around area I have accumulated snow that is quite deep (3' plus). My RL will easily drive thru this snow and the drifts around it. By EZ I mean like I'm driving on dry pavement. I touch nothing, just leave it in Drive and motor around. BTW, I'm basically driving off road here. Honestly, no surprise, as I've driven off road in the snow with every 4x4 I've owned for over 35 years. I would expect at least a similar performance from this truck.
Yesterday I decided to bring up my 18' double axle utility trailer. On the bed is 1 ton of wood pellets for my pellet stove and since my first ton is running low, I decided to move my extra ton up to the garage. The total weight of the load is 3400 lbs. (the ton plus 1400 lb. trailer weight). The snow accumulation around the area of the trailer is easily 2' plus. BUT, no ice underneath. I decided, what the hell, I've pulled that trailer out of there in years past with my Dodge Ram 4x4, why not the RL ? If I get stuck, the tractor wil pull me out with nary a wimper. I backed off the driveway thru approx 50' of unplowed snow. My wife guided me to the trailer hitch. I hooked up, disabled VSA, locked VTM-4, dropped the tranny in low (1st gear), and simply drove out of there. Didn't spin a wheel.
You know, I thought nothing of this as I've done this many times over the past years (using a couple different Dodge 4x4's). I was just satisfied that the RL performed as I thought it should.
But then I read your post. I have to believe that your rear differential is not acting as it should (I'm going on the belief that you did everything you said you did). With VTM-4 engaged, the clutch plates in the rear diffy should be locked up solid to the right and left axle shafts and both should spin together. The front diffy is different though. It is a limited slip diff and only 1 wheel would spin at a time. The only locking diff is the rear one and is activated thru the VTM-4 lock button and placing the shifter in 1, 2, or R and speeds less than 18 mph.
I only know that my wife stood by as I pulled my 3400 lbs. of trailer out of its spot (been there since last October), thru at least 50' of snow that was an average of 2' deep. When I got onto the driveway, she jumped in the cab and said "you didn't spin a wheel". I smiled knowingly and said "of course, why do you think I bought this thing ?". In reality, I was relieved. Because if it hadn't, I would have had to endure the dreaded "but your old trucks could always pull that out !".