Mukul! Who says that we're "old woofers"?
It might well be giving you improved steadiness. I have deep set eyes and a prominant brow ridge (my Neanderthal genes showing?) and use my right hand to hold the camera, thumb under the wind lever, index finger on the release, and the other three fingers pressing the camera body against my palm. The fingers on left hand usually curved but palm down with the weight of the camera resting on the knuckles and backs of the fingers unless I'm shooting a long lens. I rarely actually grip the camera with my left hand. For years I did a lot of flash photography, holding the flash off camera in my left hand. I still use a bit of bounce flash on ocasion. I guess that's how I got in the habit of the one hand hold. It's pretty much the same whether vertical or horizontal.
I've never felt comfortable shooting an SLR like that, probably because the eyepiece is in the center.
I've heard that some photographers prefer useing their "other" eye because it lets them concentrate on overall composition. The master eye seems to get hung up on looking at the details. The other eye is mostly used just for depth perception, making the world look 3-D, but the way it's connected to your brain seems a lot different. Ever try reading with your left eye? It's doable but it isn't all that easy! Even peoples' faces can be difficult to recognize.
For the past three or four years I've done most of my shooting with just my right hand holding the camera "way out there" at arm's length and pointing back at an angle to include me in the frame. Equipped with the 15mm Heliar it gets in a lot of background too. The posing and facial expressions, hairstyle, the wardrobe I pick, all pure acting...
After awhile you become adept at winding and releasing the shutter while holding the camera out there, an 1/15 of a second doesn't scare me and I can usually pull off 1/8 no problem. Longer than that and I try to brace my hand against a wall.
One thing I haven't been able to do is compose the shots as verticals shot "blind" with any certainty. I've about given up trying. Back when I was trying I was releasing the shutter with my thumb.
http://thepriceofsilver.blogspot.com has a bunch of these shots. In a few of them you can see my reflection in a window, or shadow on the ground.