I am left eye dominant and right handed. There is no problem with this that can't be cured with proper training and practice, for shooting a handgun. Don't tilt the gun. Make up your mind which eye you are going to use for the shot and position the sights in front of it. The hand movement is slight and, with practice, becomes automatic. I use my left eye shooting around the left of barricades and right eye shooting around the right.
My experience teaching people with this "problem" is to get them to stop dithering and work on effective techniques. Shooting right handed and left eye does not cause consistently shooting to the left, but improper grip and trigger control do.
Competition requires learning to shoot freestyle, strong hand, and weak hand, so one of the things you would learn to do in my class is shoot left handed, and I suspect you would find your accuracy problem is NOT caused by using your left eye.

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Science plus Art