Jack the car up under the chassis rails and check the front tyres for movement, in terms of a clock.
3 - 9 o'clock push on the tyre , this will check for play in the rack ends
6-12 o'clock push on the tyre , this will check for play in the bearings
Lift the tyre up from the bottom to check the ball joint.
Also check the steering rack bushes are solid and the column is connected tight.
Missing a rear shock???
That'd be causing it to bounce uneven and could be your problem.