When I was younger, I remember having a similar problem with SM64 Bowser. I could beat him easily, but that control stick was hard and sometimes my hand would actually blister/get red. My solution? I sold the game.(I hated it anyway...)....
So I have no real solution to offer. But using my palm... did NOT help. It made thing clumsier, more painful, pretty much worse alll around. So, I agree with SoS, use a glove. (What a way to turn a sentence into a paragraph...)