What tire should I put on my road/commuter bike to prepare for a New York winter?
You say you've got a small apartment that can't accommodate two bikes, but how about two sets of wheels?  The only thing that can deal with ice is studded tires, so for those 20% of the days that include ice, you need studded tires.  Your Tricross should be able to accommodate such wheels, so it would simply be a matter of changing wheels rather than buying a whole new bike or using transit.

As for those days it seems okay in the morning but turns into an icy day in the evening, all I can suggest is paying careful attention to the weather guessers and guessing conservatively.