I have a 2000 F250 4x4(70000mi) that I acquired about 2 months ago. On 6 different occasions when turning left as I straighten out the steering becomes very unstable and the front of the truck wobbles back and forth. Steering the truck is like trying to stand on a beach ball. When it's going straight there doesn't seem to be very much control, and when I respond to a wander left or right the steering overreacts significantly. Slowing down to a stop and reversing seems to correct the condition. After 4 occasions I had the driver's side ball joints replaced and the steering damper shock replaced. They were worn, but unfortunately the problem has occurred twice since the repair.

 

Have you looked into a new steering gear box? It could be worn out gears in the steering box, that is the only source for strange unexplained steering.

 

Double check your tie rod adjustment collars. Loose ones can make the front end over or under react.
I had a 79 f-150 that acted like that because of radius arm bushings. The wheel assembly was moving fore and aft, braking really changed the geometry of everything and there were some wild rides till I fixed it.
Also you may want to check for a loose idler arm.

bigpowerstroke may be right about a gearbox problem too. I think that it would be rare to have one fail at that mileage, but if you dont know the history, assume the worst.
Maybe the box is bad, or maybe even the steering column?
Post back if you find it.

 

It is defianetly the gear box. On my '97 it did the exact same thing. Changed every tie rod end and new ball joints, nothing. Put in a new gear box, cured the problem.

 

Check the bolt that mounts the front suspension to the frame. On my '95, after 30K miles the bolt departed the fix and the truck front end would wobble uncontrollably after hitting a bump. I had to slow to a stop to get it to stop. Might be worth looking at. Yours might just be loose.

 

Turns out it was my front right u-joint locking up. Thanks for the replies and I did go ahead and get the right/left ball joints replaced(with greasable) in the course of tracking down the problem.