There are a few things you can do.

1. Wine skin under sweater. This is a very good method. If it freezes under your clothing you've got far more serious problems than dehydration.

2. Rely on snow for your source of water, when you need it. This has the advantage on less weight carried and works well with hobbo stoves but not so well with a Kelly Kettle. Can be hard on fuel if there are other sources of water, but if snow is your primary source you might as use the just-in-time approach. Perhaps thats a poor use of that phrase

3. Corked aluminum flasks so that you can heat them if they freeze solid. Aluminum flasks can be tricky to clean, You can get these aluminum Jolt cans that work, and you could used removable foam snugglies. Not an off the shelf solution but worth playing around with, especially with the concerns over BPA and all that. I think some aluminum flasks are coated with plastic though and I'm not sure what that does. Probably not good.

I like a combination of 1 and 2 for winter, but I would still like something I can drink tea out of while hiking, that I can clean out after it having milk and honey in it. So I'm working on a system using all three.

p.s.
A hatchet can be handy in some cases for chopping through ice but is normally not worth the weight, and time. Still good to carry in winter if your in the woods and it really went super cold, like -30, or if you fell through the ice or something like that. A hatchet and a few nails and some cordage you can make a warm shelter and fire a lot faster if you ever needed to. I carry it midwinter for that reason, and because its handy for other little things even if not really needed, like tapping in a nail to hang a candle lantern or tarp tiedown, or for setting rabbit snares. If you do your winter hiking in woods that you might as well be prepared to take full advantage of it if needed. Depends on the woods. By the time you chop through you can usually have snow melted anyway. Sometimes an axe makes sense for a large group though, especially for a stationary camp. or if you want to try some ice fishing.