Sounds like it could be tendinitis. I would bet that you ride with your elbows locked. Take a good look at your position and your tension. Your elbows should be gently bent and flexible. Your shoulders should not be up to your ears. Your hands should be relaxed and you should wiggle your fingers alot.

What else do you do outside of cycling? Do you work or play on the computer alot? I had to learn to use the mouse with my left hand because I developed typing-and-mousing based tendinitis in my right arm. I have to take it easy on the typing, and I only mouse with my left hand now. I was in PT for several months to work out the issues.

BTW, they are called "hoods," but "horns" certainly is a descriptive term!