In the cold weather, air loses its ability to hold moisture, causing the problems you describe.
The solution is to add a humidifier. Either a stand-alone in the largest living area or one that attaches to the furnace. This will distrubute moisture into the air and help with your problems. Of course, you'll have to wait longer for the dishes to dry...
"If you want to make an apple pie from scratch, you must first create the universe." -Carl Sagan