The more I read this, the more I'm convinced you're actually left-dominant. It's been my experience that when you're shooting from your dominant eye the state of the other eye doesn't affect your accuracy.
Your brain only uses the information from one eye to compute angular relationships. The other is used to supplement that information for triangulation ranging.
Try shooting a left-eye only string and then repeat with both eyes open. If you miss in the same place both times that's the problem.
Or better yet, point at your light switch with your index finger. Now close your left eye. Are you still pointing at the switch?
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