sorry responses are\were so long in coming. Had an A-blot do the same thing and it drove me nuts for a while. Don't know how proficient you are at tinkering so excuse me for over explaining or over simplifying. The tightening mentioned above is in regards to the action screws. These should be tightened about about as firmly as you can with one hand using a screwdriver. Before doing that - try this. The actions screws are the two screws on the bottom of the rifle that hold the barrel\action into the stock . The one in the fore-arm is the front action screw and the one at the rear of the trigger guard is the back one. The third screw, in the middle just helps hold the trigger guard in place and doesn't do anything as far as securing the action. I would undo these screws and pull the barreled action out of the stock. Don't know about Rem synthetics but their wood stocks have two little bumps near the front of the barrel. These are pressure points that the barrel rests on. Most of the tinkering kind will sand these down completely. I would, that will free float (or just "float" the barrel) - then reattach the barreled action and see if you can easily slip a dollar bill between the barrel and stock. If you find places that are catching or even seem over close then sand those down a bit. Don't worry, you won't hurt anything. When done, a dollar should slide easily nearly all the way back. Tighten everything back down firmly and go shoot again - but when resting the rifle, try to always use the same spot to rest the rifle. I suggest as far back as possible as the stock is more rigid nearest the trigger and less likely to flex in that area. I am telling all this because my initial feeling is that contact between the stock and barrel is harming consistency. Under different temperatures and conditions tiny stresses change. Also as you shoot, the barrel heats and expands and contact from the stock is applying progressively more pressure on the barrel. This changes the harmonics of the barrel and will cause exactly what you described. Another option is to just go get a good after market stock - which you may want to do eventually. The original stocks are bought from other manufacturers and lowest bid has a lot of influence. A good synthetic stock will have either pillars, and or an aluminum bedding block. I like the aluminum bedding blocks. The new stock will be noticeably stiffer. Ones of these type start around $200 for , example a Bell and Carlson Medalist (which is what I bought for the A-bolt and it has cured all problems) to the cream of the crop by nost standards - a MacMillan for around $400. That sounds crazy but the $200 I spent made my action smoother, made it shoot all ammo better and some incredibly well, and went a long wey at reducing recoil. I figured within a few years I would forget the cash but love the rifle. So far so good. Apologies for legth of repsonse.
