Before you try anything drastic, see if changing your riding habits a bit will help. You might find that you don't have to even go as far as changing hand position often, but at least relaxing the muscles of that hand every few miles can make a big difference.
When I started putting in "serious" miles on my road bike a few years ago, I had almost the same issue, in my left hand. I realized that my right hand is busy all the time (shifting) - though it never had much activity beyond occasionally flexing a bit to flick the shift lever. My left hand, meanwhile, almost never moved (since that shifter was for the much less frequently used front derailleur.)
I started flexing my left hand, or at least letting it relax, every few miles. Soon, it became automatic. I haven't had a single problem since, several thousand miles later - in that time, I've switched bikes, switched to less padded gloves, and even put in miles without gloves.