Good post.Just to eliminate the easiest ... what pressure are you running in the tires?
Dry steering test (with video) would help as well.
How are the control arm bushings?
Steering stabilizer can be replaced after the issue is sorted.
In addition to all posts-
if no movement and it feels loose, I’d check caster, it makes steering feel light if low . It doesn’t create looseness, but a light wheel feels floaty.
Check/set the toe
definetely check the front hub unit bearings ..I had some so bad toe could not be set
control arm bushings can add drama ..rear steer