Get snow tires to drive in snow. The stock all seasons are mediocre for snow. If you want to replace them with other all-seasons, consider Michelin CrossClimate2(assuming they are made in the right size for the 3.
Stock mxm4 are horrible in the snow. I have run the oem mxm4, vikingcontact 7 (similar to your r3s) and crossclimate2 on rwd model 3. I also run r3 on my subaru.
The model 3 rwd traction control is great and super quick to respond. I think you and several other responses here are right, you were probably pushing slush around causing some side motion. There isn't much to be done about that, just drive slower. This is true in any car.
I've driven the viking contact 7 on my rwd model 3 through some heavy stuff and it's been really great. I've had no issues. The heavy car also help to push slush away. Only minor complaint is the low ground clearance, fairly easy to get hung up.
I would not drive on the mxm4 in snow or ice again. They are bad.
I've switched to cc2 for year round use. I really like not swapping wheels twice yearly. They aren't dedicated snow tires but gives better performance in most conditions (dedicated snow tires on the freak 70f days in winter is costly and almost dangerous) and more than adequate in snow. With that said... if I didn't have the subaru with r3 snow tires, I'd still run dedicated snow tires on my model 3.
I'm in the Philly area for reference when I say snow/ ice.