A hard rear bag that does not allow for the sand to relocate inside the bag. By either recoil and the disturbed rifle movement in the bags after each shot takes place can be a big help. This would also hold true if moving down to another target as the rifle does not have to re-settle the sand back down in the rear bag.

Thanks everyone for your help. In the last few days I have made some changes to my setup. 
I am shooting a Win. Highwall with large Swiss Buttplate. The knob on the lower prong of the buttplate rides the groove between the rear bag ears so that the stock can recoil straight back. 
I am working with a Caldwell BR front rest which is not the best but I am happy with it overall. I elevated the main column as high as it will go to minimize the amount that I have to use the elevation screw. Doing so has greatly reduced the amount of wobble in the elevation process.
With a barrel sled the rifle can recoil more or less straight back and come back to battery. I have a rear bag filled with heavy sand which was supposed to be level across the top but has a very slight slope to it. When shooting warm ups and sighters, I need to adjust elevation for the first several shots as the butt settles in. I shoot at least ten sighters minimum before moving to the score targets. 
I also use a foam pad to absorb recoil so that I can put minimum forward pressure with my shoulder on the buttstock, to make it easier to get consistent pressure. Shooting a 38-55 in free recoil with a 12 lb rifle is NOT an option. I believe that getting consistent forward pressure will result in more consistent recoil and less vertical stringing but time will tell if this holds true in the long run.  The last two times I shot I did NOT get any first  hits  outside the 24 ring ( when moving from one target to the next) which is a huge improvement for me. 
I also check return to battery to make sure its consistent. 
I will make it a point to try to settle the buttstock by lightly tapping it with the palm of my hand in order to see if I can minimize the amount of vertical adjustment I need to do while warming up/sighting in. Not sure if that will work or not. Last two times out I shot 246s on each of the score targets when moving from the spotter target. There is lots of room for improvement but if I can do that consistently and maybe a little better on occasion I will be very satisfied, at least for the near future. 
Any other ideas will be greatly appreciated. 
Thanks!
Joe S