Sounds like dyshidrotic eczema aka dyshidrosis to me. I had that for years. I'd get slightly different effects depending on precisely what type of skin it was on... finger tips, sides of fingers, tops, palm, sides of hands... also it was on my toes a lot. I'd get tiny dots under the skin, sometimes in clusters, sometimes red bumps like pimples, and they'd burst and leave 'dry skin' and peeling. It could be itchy as hell and also made stuff like washing dishes hellish, like my hands were covered in papercuts. My hands at times itched like mad, to where I'd practically strip the skin off scratching them, and that would help for about 1.5 seconds. It would sometimes go away for a week or two and then come back, but was present 95% of the time.
It was misdiagnosed by PCPs as athlete's foot several times, even when it was on my hands, which didn't make much sense but doctors were "oh yeah! it can be on your hands!". But my hands were quite dry. The prescription steroid/antifungal always cleared it up and it would go away for months, but OTC antifungal creams and sprays did nothing. Hmm.
Anyway, I finally figured out it def wasn't athlete's foot and saw a dermatologist. He said dyshidrosis immediately. I'd suggest to see a derma if possible. Anyway, if it is that, the treatment, other than temporary abatement from steroids, is figuring out what the trigger is. It's a pretty common autoimmune condition with 'unknown triggers', which I interpret as meaning it's different for various people. For me, it turned out to be gluten. I had to go GF for Celiac disease and my hand & foot skin issue cleared up entirely for the first time in 20 years.