This will be a long response as it is about 2 separate conditions.
Firstly, I suffer from extremely cold feet but excessive sweating on my face and scalp. I have been diagnosed with limited CREST syndrome approx 15 years ago, it was mainly the Reynauds part that affected my hands. The head sweating I put down to hormones/menopause/ my father was the same. I have bought so many personal fans (hand-held, round the neck etc) and sometimes before going out I would put a wet cloth on the back of my neck, or tried rubbing peppermint oil back of neck, tried "face deodorants ". I finally did some research and found the name of my condition (craniofacial hyperhidrosis) and it can be treated with ditropan. I was prescribed Oxybutynin and it has changed my life!
But back to the cold feet at night (even on hot nights, I live in Australia)- I thought it was the Reynauds and started soaking my feet in warm water before going to bed, then started wearing socks, then needed heat packs at my feet, then progressed to heated socks (fabulous!) And then ugg boots as well. I was diagnosed with peripheral neuropathy about 3 years ago, at first the Neurologist thought it could be attributed to the chemotherapy I had in 2018. They consulted with another Neurologist and did a range of blood tests and discovered my body was making an additional protein, so I was also referred to a haematologist. The medical term is 'highly positive paraproteinaemic (anti-MAG) neuropathy' . Between the 3 of them they came up with a treatment of an infusion of rituximab. After more blood tests, the conclusion is that my condition is not 'better ', but it hasn't worsened. This is all I can hope for- to stop it getting worse. So I have 6-monthly blood tests and seeing the Neurologist. And yes, I still need my heated socks and ugg boots! But I was impressed with the initial Neurologist, who was determined to find a treatment- even doing research on their own time and calling me at home on a Sunday night! The Haematologist was also very invested, and found an article about "Efficacy of rituximab in anti‐myelin‐associated glycoprotein demyelinating polyneuropathy: Clinical, hematological and neurophysiological correlations during 2 years of follow‐up...", in Wiley Online Library. Unfortunately I can't post a link (being a new member of this group) but you can google it.