Sounds to me like you have a problem with your 4Matic system. Have you checked to be sure there is hydraulic fluid in the resevior? Does the dashboard 4Matic light stay on after start up? Has the system been disengaged by the switch in the engine compartment?
We have a '92 4Matic wagon, and it performs oustandingly in the snow! We have never experienced any of the symptoms you have described, and find it superb in slick weather. Perhaps there was a layer of unseen ice under the snow each time? That could cause the actions you described. We use Nokian NRW's, (now sold as WR's), on all four wheels in the winter, (and Pirelli P-400's in summer), and drive to northern Vermont twice a month in winter and have yet to get stuck, and rarely even see the 4Matic light come on, (which indicates that the system is engaged). Plus, at low speeds, (up to 12 mph), the 4Matic system should engage automatically, (without the light coming on), so when you are experiencing your symptoms, the 4 Matic should be working.
Perhaps one of the sensors is defective so the differential has locked, thus causing the plowing? I'm no expert, but from my experience, your car should go through anything your winter weather can throw at you. It should certainly outperform your Mazda! I would take it to a good tech for an immediate look over ASAP.
1962 MB 220SEb "Fintail" Sedan
1963 Vespa VNB 4T Motor Scooter
1992 MB 300TE 4Matic Wagon
1995 Subaru Legacy AWD Wagon