First off, Happy Easter to everyone.
When you say digging in, are you resting the barrel on the bags or the stock? I actually shoot the best off a bi-pod and rear squeeze bag I built myself. I couldn't find a bag built the way I wanted it so I went the fabric store and put one together and filled it with rice. My first attempt wasn't that great do to my lack of sewing skills so my wife bailed me out on the second bag.
Depending on the stock and barrel, some actually shoot better being bedded the full length of the stock rather than floated. 90% of our work is a "standard" bed and float whereas the rear of the action is bedded, the front, recoil lug and then out to about 1" forward of the chamber. I'm not saying this is the best way or the only way, but I am saying we have the best results doing it this way. Depending on the stock and action, it's quite common not to bed the rear of the action as well. I have two personal rifle my Grandfather built in the 50's that are full length Manlicher stocks (270 win on a 98K and a 308 NormaMag on a 03A3) that both improved by large amounts by doing a full length bedding.
On another note crowns are another huge place the gun can improve. I've taken a few guns that shot in the 1.5-2" range with what appeared to be a good crown, recrowned them and they tightened up to .5". Outside of physical damage or a rolled crown edge it's hard to see a bad crown without truing the bore up in a lathe and spinning it up and watching it. Again, this is just our way of doing things, but we center the bore to within 0.0005" run-out of true and cut a 90 degree crown with a target recess.