I have a few comments/suggestions from my experience.
1. I would get a steering stabilizer (look at a place like So-Cal speed shop), they're an easy install and will help the bump steer a lot with a straight axle.
2. As you mentioned, try getting a set of narrower tires to try going for a drive. The wider tires will tend to track more and cause issues like you mentioned. My 2004 Avalanche has the same size tires, and they tend to track ruts, even with modern suspension. I can't imagine how scary this could be on your 53 at times.
3. I wouldn't expect the sway bar to make a change in the tracking. Has the ride height of the truck been modified at all, or is the only modification the wheels and tires?
