From your comments, you are not a stranger to the challenges of shooting at long distances and obviously are accustomed to shooting for accuracy.

Your 7mm Rem mag is a very light, high recoil rifle that is meant for hunting, not precision shooting.
Long range precision rifles are heavy for a reason - weight mitigates some of the recoil and stock muscling that really impacts accuracy, especially at long distance.

Light rifles are very hard to shoot accurately unless you really work on your set up and stock handling technique.
They are light to make them easier to carry in the field but that is a compromise that doesn't favor accuracy.
From your comments, you already know that.

Hunting ammo varies a lot and, if you are using budget ammo, it may be influencing the accuracy.
Several of my hunting rifles have very strong bullet weight and shape preferences when I try to shoot them for accuracy, but they do their job when hunting beautifully.
With a 7mm Rem Magnum, target game wouldn't know that the bullet POI was an inch off your POA.
The kinetic energy of that round is enormous and overcomes 1 inch of inaccuracy.
At the range when shooting targets, the light weight shows in the POI at long range.

Choice of LH or RH is entirely a personal one and depends upon your eye dominance and trigger dexterity.
I am left-eye dominant but I shoot right handed because I have much better trigger control with my right hand.
I can see the target in a scope equally well with either eye, thanks to the rear reticle focus feature on my scopes.
With a scope, eye dominance is not a big a deal for me because I close my left eye so trigger control, which effects accuracy results a lot, made the decision for me.

You might put the rifle in a solid rest and see if the accuracy improves.
If it does, you will have to address your light rifle shooting technique.