Similar issue here, some suggestions:
1. Try foam grips
2. Try Ergon or other similar grips
3. Strengthen your hand using a Powerball or rock climbing finger strengthening gizmo
4. Change your shifters, I was getting problems operating the front mech in particular, so went to a grip shift
5. Fork set up - more negative pressure, try a more supple coil fork
6. Wider bars seem to have helped me a bit
7. Make sure your gloves are big enough and not restricting your thumb movement
8. Core strength - rely less on your hands/arms to stay upright.
9. Any possibility of arthritis - maybe get an x-ray
10. Brake reach adjustment and angle
It's a bugger isnt it... I dont think there's a quick cure after seeing a very good sports physio. I have the most problems on fast, fairly smooth, trail centre type routes. It's no where near as bad on slow, techie stuff fortunately.
Probably not an ulna issue, not where you've described the pain