I sweat...a lot. My feet sweat so bad that if I don't use Drysol (prescription strength anti-perspirant 20-30 times stronger than over the counter stuff) to prevent them from sweating the sweat glands rupture due to over activity (Then I have to the other prescription for Cortizone cream to keep my feet from cracking and bleeding). All my life I struggled with my feet freezing in even mildly cold temperatures...it's one of th reasons I gave up skiing when I was younger. Managing the sweat helped an amazing amount.
Not at all related, but you can ask your doctor about Botox injections. They've been used lately for sweaty palms in folks with Social Anxiety... no reason why it couldn't be applied to feet, AFAIK.