I just happened to run across your post here this morning as I was having my coffee and couldn't help but smile.
I worked on a friends Remington 700 AAC-SD 700 in .308 for a month trying to eliminate some inconsistencies, one of which was vertical stringing even with custom, ladder tested handloads.
The easy solution and one that I suspect you may have already considered but ruled out due to budget or other considerations is get a better stock than the "tupperware" one you have.
Having said that, here is the only way I was able to wring out consistently good groups with his rifle and my handloads.
- no bi-pod, use a front and rear bag. Make sure to put the front bag as close to the receiver as possible and make sure it is in the same spot every shot.
- the rifle does not like a "free-recoil" style of shooting. I had to pull the rifle back into my shoulder with consistently good form before I started getting consistently good groups.
- if you don't reload and factory ammo is all you can get than you're stuck with limited choices. If you reload, try some bullets with a longer bearing surface. At short range, I had great luck with Sierra ProHunters.
Still,after having said all that, really the only really good way to get better groups is to get a good wood or composite stock, free-float the barrel and bed the action. Once there, you will still have some issues but they will be much smaller!