Re: cold bore accuracy
The cold bore shot is, of course, the most important one while hunting but I can see why it would be irritating to have the impact point shift. I have a rifle that puts 2 or 3 in the right place before groups walk but that's a little easier t live with. With my rifle, it also doesn't shift the impact point as much as what you're having. Some observations:
1. I have a hard time believing that one, two or even 3 shots taken without barrel cool downs would be sufficient to heat a barrel enough to warp it. In my opinion, something else may be going on like perhaps heating of the bore making it slightly smaller which could mean higher pressure and velocity which would cause a lower group. If it is barrel heating and warping of the barrel, it's possible that cryo treatment of the barrel would help.
2. You say your groups are from a fouled barrel so I assume there is no oil in the barrel. Oil in a barrel will cause the first few shots to be high.
3. I doubt bedding is the issue but it could be that the barrel is pressing on the stock somewhere where it shouldn't be.