Thanks for all the great advice... Very helpful.
To answer some basic questions: Yes I use a sling. Yes I shoot right handed, and I am right eye dominant. The rifles in question are not dedicated target rifles, but a pair of bolt sporters used for hunting. My 243 is shown attached.
I positioned my scope based on what seemed natural when shooting off-hand, and this same scope position seems to work well from bench, sitting, and kneeling. But perhaps it should be more forward?
Kraig, I think I am beginning to understand that I have been shooting prone position incorrectly my whole life. I have always been as accurate from a sitting position as from prone, sometimes even more so. And my prone shooting is not as good as when I shoot from a bipod or sandbag at a bench. Isn't it true that most people shoot prone better than sitting, and about as good as from a bipod?
If I get into position, and then I relax, several things happen. (1) the sideways pull of the sling on the bottom of the fore-stock rotates the rifle about the barrel axis, so now the reticle is not horizontal, but canted about 10 to 15 degrees (obviously not good). (2) If I relax, my head falls forward and takes my eye away from the scope.
This tells me a couple of things (1) I am using the muscle power of my right hand to hold the rifle level and uncanted (2) I am using muscle power to hold my head up and to maintain a cheek weld. So obviously I need to change something.
I know this kind of thing is hard to diagnose over the internet, but any additional thoughts would be appreciated.