I would say check the bottom ball joints. If they are a loose as that sounds then that can give you the shaking.
You could try by getting someone to shake the steering whilst you grab hold of the ball joint, or raise the wheel off the ground, grab the wheel at 12 and 6 O'clock positions and wiggle it, if there is movement then chances are that it's the bottom ball joint (you've ruled out the track rod ends already). You might also feel movement/knocking shaking at 3 and 9 o'clock too being that loose. (If you have a helper then they can shake/wiggle while you feel the joints.