Congratulations. Sounds like barrel timing was the main issue. Now that you've found it, try the other seating depths over again, if you haven't already. Until you get down to a reasonably tight group size, their effect can be difficult to discern.
After that, concentricity would be the next thing to tackle, as it is also more important to secant ogives than to tangent ogives. The reason is the tangent ogive starts with a zero angle of departure from the bearing surface of the bullet and it gets larger from there. As a result, some part of the tangent ogive always touches the leade angle into the rifling in the throat before the bearing surface does, so you tend to see some self-centering as that radius centers in the conically tapered leade. The secant ogive has an immediate angle of departure at the shoulder with the bearing surface, so the leading edge of the bearing surface usually makes first contact with the rifling at the groove diameter, so rifling marks start to be extruded into it before centering in the cone can occur. The centering forces are then not great enough to eliminate any tilt the bullet already has. The only cure is to have the bullet straight in the first place.
One thing you can do is get a Lyman M-die and set it up to expand only enough to put the small leading step into the case mouth (not the flare portion). If the cases will seat that way in your chamber without using a crimp die to bring the step back down again, the slight widening can actually help center the case mouth in the neck portion of the chamber. But even if it doesn't, if you don't have either the Forster or Redding competition/benchrest seating die, both of which tend to minimize runout, the step in the case mouth lets you seat a bullet upright in the case mouth so it starts straight into the seating die, and this cuts runout considerably. You already are using a good method to keep runout out of the case neck, so, between the two, you should get pretty straight ammunition.
