"Yes, the Savage 340 has that reputation for being "shifty" in all calibers, probably due to the barrel band which holds the barrel to the forend. There is no easy fix for that, it's a design flaw."
You are very right. Worked on a couple of Savage 340s over the years. Eliminated the barrel band and tried to glass bed the rifle. Tried all kinds of other stuff, nothing worked. Finally just gave up.