Hi D.!
I'm a doctoral candidate in cosmetic science, so I hope this helps. You need to stop using soap (that is, anything that says "soap" somewhere in the name). It is way too harsh for your skin, especially in the winter when it's less humid outside. Try Cetaphil cleanser on the face, and even the body. You could also try one of the Olay Quench products (for body), they're designed for African-American skin, even though that's not how they're marketed. Or look for one of those creamy, moisturizing body washes. No clear cleansers, they are too drying. Look on the ingredient labels for petrolatum, safflower or sunflower oil (for body products), and for cetyl, stearyl or cetearyl alcohol in face care products. At least one of these should be near the top of the ingredient list.
Also, if you're not experiencing acne, you could certainly cut back your face washing to once a day in the evening and then just splash with some warm water in the morning and moisturize again. If the burning that you are experiencing when you apply moisturizer lasts more than a few minutes, you might try a fragrance-free moisturizer. Olay's Total Effects comes in a fragrance-free version and is pretty rich but won't cause acne.
Good luck!