perhaps some heavier, non-bike gloves are in order?
There is winter gear for cycling. I use Pearl Izumi's AmFIB gloves and my hands are sweating. Add to that tights that have a windblocking lining along with some good thermal underwear, a gore-text jacket with a thermal turtleneck and winter cycling shoes and I tend to be sweating and drench by the time I get to work (and that's in 3 degree Celcius or about 37 F weather). Remember, layering is a good thing. Might want to visit the Winter Cycling section here.