I have used (not very freezy here) wide mtb tires, lower psi front..low psi -both really.
I keep off the saddle and weight whichever tire need the weight\pressure.
Front slippage is a hard correction, keep the bike very upright. I've done corrections but can't really explain how...I learned it mtbing. Maybe moving the frame into the direction of the stee\uncontrolled drift. Moving a frame sideways means sliding the ony tire in traction.
Go with the flow.
But it can work, you also can strain the mucles in your inner thighs moving the frame sideways rear.
The rear slipping is not so hard eh?
Drop a bike the same as a motorcycle, rear brake clamped -slide it sideways and get off. The leg to the underside -( you drop the left side of the cycle...why?) you lift left first! Ride it out skidding with on\with the bike.
Bad stuff, get the bike away from you if you can, chuck it.
Bikes can hurt.
Sorry to hear about the fall. Some ice you don't see. Keep very upright cornering.