I had the same issue a few years back when I started riding long distances on my fairly upright bike with straight handlebars. The good ladies here at TE all strongly recommended a proper bike fitting. The bike fitter re-built my handlebar/headset, changing out a straight bar for one with slightly angled ends. This made all the difference. I also have bar-ends which allow me even more options for positioning my hands, so I can move them around more on long rides.

Good luck!