I shoot with both eyes open but I suspect that it may not be your eyes causing the problem. The problem might be your mouth is open. When shooting at targets you don't have any adrenalin flowing, you are breathing normally and fairly relaxed. Put a deer in front of you/me/anyone, your heart starts to race, you start breathing heavy and your mouth drops wide open to suck up more air. When your jaw drops, your anchor point gets screwed up and all of a sudden your peep isn't lined up and everything is out of wack.
This has happened to me, I drew on a deer and it felt like I had someone else's bow, nothing felt right until I figured out to close my mouth and breath through my nose. Just one of those things that is difficult to practice and something to be aware of when out hunting.