In addition to the above, try double wrapping your handlebar, or use a shock/vibration resistant wrap.
The cheapest wrap that reduces road vibration is an old tube, cut to strips shaped roughly like bar wrap. Wrap the tube around the bare handlebar, then cover it with whatever cosmetic bar wrap you like. Butyl inner tubes contain carbon black that will smudge everything.
Or try something like Arundel Synth Gecko, a 100% silicon rubber wrap that's soft, thick and very resistant to vibration. It's pricey at around $30, but a great value because it can be reused indefinitely with reasonable care. It doesn't use or need any adhesive -- the wrap adheres lightly to itself and to the bar. It doesn't take a set or permanent curl so re-wrapping is non-critical -- after removing from the bar let the Synth Gecko sit or hang awhile and it'll straighten out.
I double wrapped the bar on my carbon fiber bike this summer with cheap "cork" foam tape on the bar and Synth Gecko around it. Very comfy but I took off the Synth Gecko a couple of weeks ago to borrow for another bike. Since then I've had some minor but noticeable aching in my right thumb joint and base of the palm. The extra wrap really did help.
There are also compression gloves that some users report offer some comfort from strain, arthritis, etc. My only concern about compression gloves would be winter use. Sometimes compression can hinder blood flow and make our fingers and toes more vulnerable to stinging, numbness and even frostbite. We usually need a little air space for efficient body heat and tight gloves thwart that. But for warm weather use it should be fine.