My first advise to you has been already mentioned; get a good bike fit done. My second is to relax your shoulders when you ride. This will cause your elbows to bend and keep the pressure off the bars. You don't need a death grip on the bars to ride, you only need to have enough pressure on them to steer. If you find your shoulders are shrugged up when riding, I'll be willing to bet that your arms are stiff as well. Relax your shoulders, bend your arms and get a good pair of gloves. Several have mentioned the Specialized gloves. I use their BG Pro series gloves and they are very nice. If you have drop bars with STI shifters, ride on the hoods. That also helps bend the elbows.
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HCFR Cycling Team
Ride Safe ... Ride Hard ... Ride Daily
2017 Colnago C-RS
2012 Colnago Ace
2010 Giant Cypress