When shooting quickly at close to mid-range targets I shoot with both eyes open and focus on the spot on the target where I want the bullet to land.
I am right handed and right eye dominant, but if I think about it there are two sets of sights when I do this. It really doesn't matter, if you get either set of sights lined up on the target you will get the hit. For me, the extra set of sights to the right are the left eye set of sights.
After enough practice you will easily be able to ignore the "wrong" set. Your only risk with this method is trying to line up the rear sight on one set with the front sight of the other set. Of course if your arms are of normal length the gun will be nearly sideways. Unless your eyes are too close together.
The only time I focus on sights is for lining up very long shots; whether for pistol or rifle. I apply the same principle for practical rifle and pistol.
I drive a car the same way. I suppose some people focus on the hood ornament or left front fender and get around that way. I focus down the road on the point I want the car to pass through. I use peripheral vision to line myself up with that target.
Focus on a spot on the target, bring the gun up through your peripheral vision into your line of sight until one of the sight sets is on the spot then squeeze.
It was odd at first, but when I'm "on" I don't actually remember lining up sights on the targets, I only remember focusing on the impact points and seeing the holes appear. Give it a try.