Hmm, that sounds like something is loose/worn with the steering rack - mounting or internally. You mentioned something about play with the mounting bolts? This should be addressed first - the rack needs to be rigidly mounted with zero play. Any movement of the rack in the mounts will translate in side-to-side uncontrolled motion (called death wobble).
Believe these are 1/2 inch diameter bolts (grade 5) in the kit and they can be tightened to up to 65 ft.lbs. if it gets everything rigid.
If the steering rack is tight and this persists, then the gears in the rack have worn and this will likely require finding another rack. Note that many GM models used this same or similar rack.