Ride without your gloves to see if that helps. If it does... Problem solved.
If not the take note of where you put your hands the most (hoods, drops etc) and make a point of not riding there at all and see what the result is. Try the other two common positions - ie if you ride on the brake hoods, ride on the drops for a while (raise stem if required) and ride until you notice problems. On your next ride try the other bar position.
You could have a nerve or a blood flow issue. While the latter is uncommon its not unheard of. You wouldnt be the first person to tighten up there gloves too much so thag when you angle your wrists when riding blood flow is reduced.
Do you have your saddle positioned fairly far forward on the seatpost? You could be putting too much weight on your hands. I personally find that a more stretched out position suits me better than a shorter but lower bar setup.
Put your bars somewhere such that you can ride on the drops. It's a great starting point - not the brake hoods. A good position on the drops will naturally place the wrists straighter and bent arms will absorb road shock better.
See a good,'well recommended' physio. If you're in NorCal i cant help you with that one...