I've always found it tough to get in a good amount of practice with centerfire... but if you can shoot a good break barrel air rifle (or a .22) well, you will have the fundamentals of marksmanship. At least with an air rifle you can generally practice at home with appropriate safety backstop and targets. The reality is... whatever you shoot on a bench on paper will your BEST shooting, compared to a shot on the field, a shaky rest and your heart pumping.
"Go slow to go fast" - make sure you are a decent shot with an air rifle or a .22 in different positions, before spending $5 a pop on factory 6.5 CM ammo. The last thing you want to do is wound game because of a poor shot.
As for your setup and eye dominance, you will also want to make sure the reticule is focused (adjust the ring closest to your eye, not the zoom ring) and watch for parallax error where an inconsistent head position will move the cross hair. Some scopes have parallax adjustment, some don't... so best way to minimise is to get as consistent cheek weld as possible and not swap aiming eyes.
For the first shots from that rifle, and the first shots for some time with centerfire and an unusual shooting situation at a range not a bad effort.
My advice is go with the cheaper option either air rifle or .22 in as similar a configuration as you can get to the centerfire and find someone who's happy to give you a bit of time mentoring or coaching your technique. I see a few possible technique things that might be going on apart from the identified double grouping from changing eyes and shoulders but I can't really confirm until I see you shooting or I could well make your results worse and just confuse you...
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