Dont grip the bars hard.
Try and ride with your wrists straight, not bent. Bent wrists tend to concentrate stress rather than transfer it. Elbows should be bent to provide some suspension, and shoulders should be relaxed and dropped down, not hunched up.
Get your bars in a comfortable position. Play around with your brake lever position, up and down the bars, and around. I like a little bit if inward cant in my levers.
I also prefer slightly narrower bars than usual, with a small radius drop. Wide bars are often recomended for comfort, but many people ride with their arms further apart than their shoulders. You will never get a good shock absorbing position if your arms are triangulated outwards.
Getting a good position involves a lot of subtle body awareness. Try taking up an excercise like Tai Chi, Alexander technique, yoga etc.