Most '06's will shoot 150gr and even 125gr well, but then some just won't. My Brother has an M-1 that shoots 125gr Sierra's w/4895 VERY well (it has a 1-12" twist barrel). He uses this for the 200yd "Rapid-fire" stage of NRA bullseye competition. Very successfully too.
I suggest you try going to the 165gr bullet though; specifically, the 165gr Ballistic Tip.
Also, try IMR4350; specifically 56.0gr for about 2775fps from a 24"bbl.
This is the load I use for "trouble shooting" an '06.
If it dosen't shoot well with this combination at varied seating depths, someting IS wrong with it.
Most good bolt actions will shoot at or under 1", 5-shots at 100yds with this load.
I have a Rem 742 I bought as a "basket-case" for $100. After extensive cleaning, disassembly, tweaking,re-assembly, proper torqueing, ect.......... It will shoot this load to 1" for first 3-shots.
Some '06's prefer even heavier yet though, the 180gr bullets.
I use 55.0gr of IMR4350 or 58.5gr of RL-19 for 2,650fps. Shoots almost as flat as the 150 and 165gr bullet, but LOTS more thump! (Both ends!).
Or, try 59.0gr of H4831SC or 60.0gr of RL-22 for even more thump! (2,750fps!). However, best accuracy w/ Rl-22 is at 58.0gr. Never shot an an '06 with this load that didn't shoot it well.
(Warning: Don't use these 165 or 180gr loads in a M-1 Garand, they'll eventually bend your op-rod)