I run a snake through the barrel every time, after each shooting session.
Based on your symptoms, I would think you have a pretty fouled bore.
Boresnakes don't do well on heavily deposited copper. They are fine for light carbon buildup, but you need a copper removing solvent to really get a clean bore.
I would suggest:
1. Deep cleaning the barrel with a copper removing solvent. You'll know all the copper is out when the patches don't turn blue. You may end up with forearms that look like Popeye's, but the gun will never group with heavily deposited copper in it.
2. Re-checking the torque on the rings. If you have access to a torque wrench, it will be all the better.
3. Use better ammo. I have never had what I would consider a decent group with surplus FMJ's. The Federal Premium Sierra Matchkings you listed should produce a decent group.
If all else fails, go ahead and bed the action. It won't hurt anything, and almost always makes some improvement.