This has come up before, recently. Try searching for wrist numbness, or tingling, or ulnar nerve. The bottom line is that your body is protesting being in an unnatural position. Don't ignore the problem!

It may not be just your handlebar type; your bars could be too far from from your seat, and/or too low in relation to your seat. Since your hands are suffering unequal effects, either your bike or your body may be asymmetrical.

Gloves with padding in the wrong place can also be a problem. I like Specialized Body Geometry gloves, but a guy working at a local bike shop finds that Specialized Body Geometry gloves cause discomfort for him, because the edge of the rather prominent padding is right over a nerve.