"What do you do with the moisture that gets trapped in your bag overnight? I can see airing it out on my pack sometimes, but not most days. Doesn't your bag end up moist, and less effective?"
This may sound odd, but don't trap so much moisture. Don't cinch up your bag too warmly, and that lets some steam out via the head opening.
Yes, the bag still accumulates some humidity. When you get up in the morning, there will be some finite amount of time before you are packing up to move. So, as soon as you vacate the bag, you turn it inside-out and try to hang it up someplace dry. That might be in the tent, on a tree branch, or between two skis in the sunshine. You try to let it steam dry as much as possible before you stuff it.
"Are you not in a tent, in that case? Tents are problematic because of all the trapped moisture if you don't have good ventilation."
You just answered your own question. Ventilate the tent better. You are not supposed to seal up the tent to preserve warmth. Instead, you close it up enough to shed most wind and precipitation, but you maintain some airflow to keep things semi-dry.
Sometimes I am in a tent on the snow, and sometimes I am in a snow shelter down in the snow. Strong wind blowing through is bad. A bit of ventilation is good.
I would not recommend the use of a woodburning stove in a snow shelter, but often we will run a white gas stove in the doorway or vestibule. I've tented as high as 19,500 feet elevation with the white gas stove burning in a vestibule.