I live in the DC area also and, happily, just picked up my P85D with 19" all seasons. We had about 3 - 4 inches of snow on the roads and lawns that achaar mentioned... but it was unexpected so no plows and lots of inept drivers. Now...
I ended driving my recently hand-washed by me in the cold (over the weekend) D into work in all the mess and snow. It handled it flawlessly. I won't get into the winter tires (which really do make a difference) vs. all wheels debate but all I can say the car handled it all with ease.
By comparison, my last car was a 2013 MB S550 RWD only with AMG rims and summer tires. It was awful in just a dusting... truly a sled. Could barely pull out of my driveway, which is flat!
I realize the OP's issue was a pulling to one side but my experience with my MB was it sucked, no matter how little snow there was. it pulled, slid, skidded, didn't stop. All I could do is park it...
Punchline- try winter tires on the rear and see if that makes a difference. I don't think all seasons will improve things by much. And of course there is the new D but it is a lot more expensive than swapping out a set of snow tires for the summers.
