perhaps there are two things you can work on without changing anything on the bike?
1. Make a conscious effort to bend your elbows. They are your "suspension" for your body. Think of yourself as floating over the bike, connected to it by springy elbows and springy knees.
2. Loosen your grip on the bars. (springy bent elbows will help with this) You control the bike more with your body than with your hands, and once your body gets the idea it will become easier to relax the automatic death-grip on the bars. Elbows will take the shock, so hands don't have to grip so hard.
Hands and elbows react off each other, so working on one will help with the other even if you can't keep track of both at once.
It is likely that your left forearm is stronger cuz you are a lefty, so it didn't get as sore as your right. Your left arm probably has more practice gripping things.
"If Americans want to live the American Dream, they should go to Denmark." - Richard Wilkinson