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What I originally thought was just a really bad wonder, now I see a pattern.
When I get on the skinny pedal, my 2000 F350 (turbo-7.3 and a manual trans) pulls to the left. Hard enough that if I'm not watching what I'm doing, I'll "accidentally" change lanes.
When I left off, it pull to the right just as hard.
Driving like a semi-normal human it is not as noticeable.
I'm stumped as to what it could be. I know the shocks need to be replaced but I can't just throw money I don't have at it until I fix the problem by pure luck. What should I be looking for?
Torque steer can be a problem with light weight drag cars, but these SD's are far from that, so unless you a engine modification you ain't telling us about it shouldn't be that.
I would be looking at the rear axle. Are the leaf spring bushing in good shape on both sides of the truck? Are the u-bolt tight on both sides? Can you see where one side has shifted a little? Is the truck dog tracking at all ( back not following the front )? I wrecked a truck and bent the frame, the shop said it was perfect when they finished but it still did what you are discribing.
The rear diff could be acting up? Is it LS or open? It will do that with a broken axle on one side, but you should know if that has happened. Get a four wheel alignment and see what happens. That is cheaper than shocks to start of.
You may have something loose on the front or rear axle. Probably the only way to check the front it to crawl under it with a pry bar and check every link/bushing for slack. For the rear axle, you could lay on the ground (a safe distance away), have someone else sit in the truck, and they could torque-up the rear axle (hold the brakes while giving it a good bit of gas like they were going to power-brake it), while you look to see if the axle is shifting.
It sounds like something is letting the driver's side of the rear axle move fore and aft. That's where I'd start looking for something loose or broken.
good luck tomorrow, hopefully you find the problem... to me it does sound like something is sticking causing one side to move towards a certain direction.