Yea I have a feeling your tires are not up to the task in your conditions. Also, it is a good idea to mount snow tires on the thinnest rim you can find. The wider the contact patch, the more the snow will be compacted and you will not be able to dig in properly.

I'm from Russia, so lots of experience with this. Especially being a dumbass at one point and putting snow tires on a 10 inch wide rim once. That was a total waste of money. Glad I was able to unload the setup.

I bought a set on here from LeMans. Forgot the name of the tires, and didn't get to try them out yet. I had blizzaks on my last car, and they were amazing. It was a RWD Lexus GS430. Also a V8 out front, but no AWD to assist. I was able to drive the car as long as the snow wasn't over the bumper.

Also remember, your car will under-steer under braking in the snow. You really need to be turning while rolling. So if you are on the brakes in the turn, you are trying to apply 2 different forces to the tire. One to stop, and one to turn, this limits the performance, of both characteristics. The tire now doesn't have the proper grip to stop, or turn. So it will take the path of least resistance, and go straight.

Try this in a lot or something.
Slow before the turn, wheel straight, and try rolling through the turn (no brakes), apply throttle as necessary.

Just remember, turning under braking is never a good idea. You must keep you tires rolling to negotiate the turn. If you lock, or abs the wheels, you will not be turning.