I have similar symptoms. First, I ordered Grab On Grips, which are closed cell foam cylinders that slide on your naked handlebars with a little dish detergent as lubes. Then I wrapped the result with gel padded bar tape. If you don't want the larger diameter bar, I recommend just the Grab On Grips.

Another strategy is to elevate the handlebar with a stem whose angle of rise can be adjusted. Less weight will be on your hands as you will be sitting more upright. You may have to give up the aero position.