I had the exact same problem.
What worked for me...
I consciously move my hands around, and every few miles, I just shake them out.
I put Specialized 'Phat Wrap' and gel wrap bar tape on my handlebars. They use gel pads under thicker tape to reduce vibrations.
When I was using the computer (which I do for hours and hours) I used a 'smart glove' support glove that protects against carpal tunnel, but made things feel better for my issue (ulnar nerve compression is what we're talking about here) and I make sure I don't put more pressure on the nerves.
It took a few weeks for the feeling to come back, but it has and I haven't had a problem since I put the new tape on my bike and was more aware of moving my hands.
Look up 'Ulnar Nerve Compression" or "Ulnar Nerve Damage" for other opinions and solutions. Just don't give up - riding is just too much fun to avoid!
Edit! Oh - the most important things I changed - saddle and stem position! I tilted up the nose of my saddle ever-so-slightly and changed the angle of my stem so that I wasn't putting as much weight on my hands. I used a slightly shorter stem to. Do you feel like you're supporting a lot of your weight on your hands? If so, see if you can make small changes to shift some weight off your hands and back onto your saddle.