The house stays warm in general, but cold air blows when the heat is not on. There is only one large cable leading from the furnace to a switch which turns the furnace completely off, so it's not the fan switch, is it? Can I go inside the machine to turn the fan off somehow? Thanks so much for your help!

Did you check the fan switch on your thermostat. It needs to be in the auto position not on position for the blower to cycle off.