I tried out some Vaude long bike gaiters during the wettest, grimmest Hebden ride in about 5 years of riding there yesterday, I also suffer with cold feet and hands, the gaiters (worn under trousers) finally kept my feet dry, combined with waterproof socks they made a big difference.
Reading reviews I’m not sure how long they are going to last, but these seem like a no brainer and I’m gonna keep using them, you could swap the waterpoof socks for something warmer and I think you’d still have dry feet, or combine the waterproof socks with some wool ones and a large enough shoe and I can’t imagine you’d have cold feet.