I kind of agree with Roger, although I think he missed your update to say you're taking Goretex socks rather than shoes. I would carry the Goretex socks and two pairs of spare lightweight socks. All of my socks would be wool.
My system on the MIlford, with dry huts and drying rooms each night, would work like this:
One pair for hiking, wash them and hang them in the drying room each night. Either use pink, orange or pin a bright ribbon to them when you hang them up. This avoids the "Who's got my socks" debacle in the morning of all the blue, black and green sock users!
One lightweight pair for walking around and from the hut each afternoon. You could ditch this pair if you're prepared to stay in your washed out, wet socks each afternoon.
One dry pair for evenings in the hut.
If I was walking the Milford with my wife, I would DEFINITELY carry her a pair of Crocs or similar for hut shoes. These will allow you to have warm dry feet every night, which will contribute greatly to your enjoyment of the track.
Carry the Goretex socks, and wear them whenever you like, but be open to NOT wearing them. Your feet will be wet either way, since it rains about 250 days a year and there are a couple of creek crossings without bridges. My experience is that if you think you can keep your feet dry and then they get wet, it makes for a miserable days walking. However if you accept that your feet will be wet all day, but mostly warm, and dry and warm at night, you can tolerate it quite well.