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For a $1000 budget you can get a lot of rifle. I would look for a Kimber or used Sako 75. The Savages are very accurate and the new Ruger American is as well and at $350 you can spend a lot of glass which is a smart investment. My advice is buy a basic rifle and spend as much on glass as possible. High end Nikon, Leupold and Ziess are your best bets.
I'm left eye dominant too but don't have issues with scopes. I struggles for a while with shooting shotguns with both eyes open finally conditioned myself to compensate and overcome the problem.
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