For the last time that I trekked in Nepal, I took several pairs of thick synthetic or wool/synthetic socks, and I rinsed them out occasionally. Once rinsed, I hung them up to drip dry on the rods between the inner tent and outer tent wall. Unfortunately, they froze overnight.
So, in the morning I placed the frozen socks into a plastic bag and put that in the top of my backpack. The plan was to pull them out during a nice sunny lunch stop so that the drying process could continue. No sun. They stayed frozen or wet for weeks at a time.
I now have some Smart Wool socks, and they are comfortable, but thick synthetic socks might be more practical for those kinds of conditions.