Your rifle is not sighted in properly. You now find it shoots to the right, after testing it at 200 and 300. So zero the rifle properly at 100 and fire four or five shots to check zero. (Do this after considering the below*)
To answer your other question, if you are leaning into the rifle one way and then holding it back on target, then yes it probably will shoot off. You can often tell that as the rifle will also recoil away to one side.
You should find out what the Natural Point Of Aim*is, and accomodate this in your shooting, and it will solve what you are talking about. It's basic shooting technique. There will be videos on Youtube. I thought that I invented it myself, but I then found out that people have been doing it for years.
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