Wear gloves if you aren't already, make sure they're fitted properly. That'll give you extra padding.
Sounds like circulation trouble, Pins and Needles like when your leg falls asleep? If you're using straight bars are they adjusted so that your arms are bent at the elbow to support your upper body and steer? If your arms are straight it's can be easy to rely on them to carry all the weight when your back and abs should be pulling their own throughout the ride. If your arms are straight you've likely got all your upper body weight on them and in turn on your wrists and palms which if held for an extended ride will become very uncomfortable and can eventually damage your wrists.