My wife is strongly right handed but very left eye dominant and when I first gave her a rifle to shoot she tried to shoot right handed but trying to aim with her left eye. Needless to say it was very awkward looking! So she learned to shoot left handed.
It may depend on how just strongly the dominance is expressed in you eyes. Now when I go to teach a new shooter the first thing I determine is what is their dominant eye. I agree with JPRO as you can teach hands more easily than you can train the eye.
"It is wise, though, to remember above all else: rifle, caliber, scope, and even bullets notwithstanding, the most important feature of successful big game hunting is to put that bullet in the correct place, the first time!" John Jobson
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