Bigger picture, what’s your current facial skin care routine? If it includes a lot of cleansing, exfoliating, hot water, strong soap, multiple daily washings, et cetera, dial that all wa-a-a-ay back. All of those things strip away the natural oils quite effectively, which leads to that red, inflamed look in the cold. The best way to keep your skin moisturized is to keep the moisture it naturally has from escaping, and that’s 10 times more important in cold, dry climates.
Be sure to drink enough water, too. It’s deceptive, you lose a lot of water through breath in cold, dry air, so you can be dehydrated even without sweating.
That doesn’t sound like anything like a problem. We had a similar discussion on a local winter biking group, and there were some people who had issues from washing their face three or more times a day, exfoliating regularly, and such. Yikes!
Hope the hyaluronic acid helps!