+1 re. one of the Ergon grips, as pictured above. I had various 'hand pain' problems when I started riding again; really resolved when I switched to these last year.
FWIW, a couple further observations on this:

1. Yes, drop bars inherently offer varied hand positions, but you can get this (minus the drop position [even then, just drop your elbows into a head-wind!]) with a combination of a flat bar and good quality bar ends. That, + the Ergons, has proved to be what works very well for me.
2. Many, many people, much more commonly in Europe etc. than in N.A., do long-distance riding/touring on flat bars in some configuration or other. I don't buy the 'inherent on-road superiority' of drop bars over flats; it's very much a matter of personal preference/riding style, seems to me.
3. You can suffer from ulnar neuropathy (classic "cyclist's palsy") just as easily on drops as on flats.
4. One more thing to consider: OP, you don't mention your age etc., but anyone much over 30 can easily already be developing mild osteoarthritis symptoms; one of the first places this shows up is in the base joint of the thumb; if you are using rapid-fire shifters on your Coda, you MIGHT be experiencing pain from the repetitive thumb motion of rapid-fire. I went through this a few years ago, and it took a while to figure out (sports doc.) what was going on. Got so bad I couldn't ride; switched to grip-shift, problem completely gone.