That sounds very uncomfortable. I will tell you that I have a patch on my foot and have had it for about 8 years -- um hm, right? And I've had it in two spots and it becomes like a thick area of skin that itches like a mother and then peels (then it might really hurt). It comes and goes. Never have had a diagnosis. I have only mentioned it during my annual physical though and not been to a dermatologist. BUT, it's one foot and two spots. Manageable overall.
You are having this on soles of feet and palms on a regular basis and it is related to your cycle? When I was pregnant I had the terrible skin complication that some women get of itchy hands. I was miserable. I used hydrocortisone on it per my doctor's recommendation. Any topicals help? Anyway, that during pregnancy is due to hormones. I guess I think of that because you have this in a cyclic manner related to your period.
Something to explore is dyshidrotic eczema. https://www.webmd.com/skin-problems-and-treatments/eczema/dyshidrotic-eczema#1 You do seem to have the symptoms of this. Take a look at the link and see what you think.