[old_tv_nut]

Compromising purity for anything else is not possible; you can only make it worse. So set the yoke and purity rings for best purity, and leave them there.
Don't use horizontal hold to adjust centering. This will affect the timing of the color burst gate and potentially make color unstable. Horizontal hold should be set to the center of the hold range.
Consult SAMS on adjustments in the horizontal sweep section. Note that any horizontal efficiency adjustment must be set so that the horizontal output tube doesn't draw excess current.
Use an Indianhead or similar test pattern with circles to check width and linearity. Use a crosshatch pattern to check pincushion amplitude.
Vertical and horizontal adjustments are generally independent of each other.
SAMS should give an indication of what order to do these various adjustments.

Your width problem could be due to weak tubes in the H sweep section (H driver, H output, damper).

Good luck!