Turn signal switch is just a bunch of contacts, a rotary switch of sorts. When activating, they are basically self cleaning. If the grease is old and crusty it would make sense that it doesent work in the cold. But, bad grounds,loose column connector or corroded connectors can display similar symptoms.
The flasher is thermally activated,so if its doing it in both directions, its likely not the switch. Next time its stone cold,try the hazard lights.
Uses a different flasher. Process of elimination.