Fortunately, your problem is fairly easily solved: Ditch the cotton briefs. Go to a running store or an outdoor store and buy some synthetic briefs designed for outdoor activities. Patagonia already makes something like this - that company's Capilene Briefs ($17) are fine; so are Duofold's Coolmax briefs ($11), which in spite of the name are a good insulating layer. Buy a few pairs, so you're sure to have some clean ones handy.
One other step is to buy briefs with wind-repelling panels sewn over the strategic location. RoadRunner Sports, a web/catalog retailer of running shoes and apparel, sells something it calls the RRS Tech Windbrief, which are Coolmax briefs with a wind-resistant front panel. In REALLY cold conditions, you could double these up over something like the Patagonia briefs. You might also try a little heavier tights than you're wearing, or double-up the tights with a light outdoor long-underwear tight, then a Lycra running tight over that.
Stay warm, bro. Your genetic future depends on it.