Did you have your bike setup done by a pro at a bike shop? I'd have the bike fit checked first to see if there is an issue there. And if not, see your doctor to have carpal tunnel ruled out. I have it in both wrists, but it is controlled with wearing wrist braces at night (that's when it get worse).
When riding, I try to change hand positions frequently. If it gets really bad, sometimes I'll rest one hand behind my back to take the pressure off and give it a break.