Do you use gloves?? The ones that I use and recommend to my clients are the ones that have suede palms. They definitely take away some of that pressure because they give enough padding right where the weight rests. Also, stretch your palms in between sets by holding your palms up and bending your fingers back, just like you do to stretch your forearms, but you don't have to hold your arms straight out in front.
Oh, one more thing, try not to hold the weights too tight. You have to be conscious of this. I often have to remind myself of that as well. It will hurt more, as well as take the concentration off the muscle you are working a bit. HTH