I don't own a Ruger American, but I do have a Remington 783 (in .308) and a Savage 111 Trophy (in 6.5 Creed), both of which I would say are in the same economical class as the Ruger American.
I've been able to get both of my rifles to shoot sub-half MOA with handloads only. For the Remington 783 in .308, it has the same twist rate as the Ruger and 168 grain projectiles have worked very well. Varget powder is my go-to for the .308.
I start off by jumping the lands by .020" and then work on a powder charge. Once I find a good accuracy node, then I work on jumping the lands even further if necessary (in increments of .010") until I get it to shoot at least half an inch at 100 yards.
Also, double-check your barrel crown and make sure there are no scratches or otherwise any damage.