I had peeling hands when I had eczema, and my left hand was way worse than my right hand (the left arm is the one that probably has TOS).

I got the eczema under control by using hydrocortisone cream at bedtime (i would wear cotton gloves so that the cream didn't get everywhere) and then during the day I applied the neutrogena Norweigan hand cream for moisturizing.

I don't know if eczema is a possibility in your case, but if it is it's usually pretty easy to get under control (mine got better within a few days, and was pretty much gone after 2 weeks).