My repetitive stress injury from cycling - piriformis syndrome - is considerably more debilitating than a hand/wrist issue (long story). But if there's one weak link in this 'cycling is a low impact sport with few injuries' thing it's repetitive stress. It can impact low back, neck, shoulders, hands, wrists. We spend hours daily doing the same thing and if your body is injured, it can be very frustrating.

It would seem that going to a more upright riding position, shifting the weight back on your pelvis and off the handlebars might be the way to go. But only until your hand/wrist issue improves. Long term a more upright position could potentially create other problems in your pelvis/back. But you deal with these things one injury at a time. Good luck and I hope you find a way out of your pain. I'm dealing with mine a day at a time. But cycling generally helps - which is extremely ironic since cycling probably caused my chronic pain in the first place. I'd go to a doctor about this but their cluelessness on this issue is hard to describe.