I don't think a new steering box will fix that movement. Unless the bolt tabs are cracked. Save your money on the box now and work on what's wrong. Unless you know for sure the gear inside the box is worn. I doubt you could tell since the whole box is moving around.
It's the bolts are just loose or the whole K-member steering mount or welded nuts inside the K-member are messed up.
Try to tighten the bolts you have on there now. It's common for them to get loose. It's also common for the welded nuts inside the K-frame to break loose and they will spin so you can tighten the bolt properly. Look for cracks and rust around the steering box mount.
Need to go in there and spray degreaser and powerwash the area. Or just clean good with strong degreaser and small brush (old toothbrush) That way you can check for cracks and stuff. And work on it easier.
Have you inspect the driver side lower control arm bushing? They commonly go out because power steering fluid will leak on them.
You need to inspect every joint in the steering system closely. Do just what you and your dad did with the box. Jack up the car and move the suspension or steering slowly back and forth while someone looks at the joint eye to eye and even feeling it with you hand over the two joints. Do it from the steering wheel and then by moving the tires side to side by hand.