In reply to Guy:
Have you tried waterproof socks for running? I find Sealskinz start to leak almost immediately, the membrane is quite fragile, but even when they do let a bit of water in they still prevent the cold water from 'flushing' through and help to keep feet warmer.
I have a friend who runs in neoprene socks (meant for surfing/kayaking/sailing) and swears by that. They also don't keep feet dry, but help with warmth by preventing the cold water flushing through.
Just lately though, I've gone a step further and started wearing Reed Aquatherm socks over thinnish woolly ones. It's like a slightly more sophisticated and hard wearing version of bread bags over your socks - they're completely waterproof and not at all breathable. Consequently feet do get a bit sweaty over time. I only run for a maximum of an hour or two at a time though, so that's not a problem.
https://www.chillcheater.com/gloves-and-footwear/aquatherm-socks