I do NOT have a rear radius arm.... I have leaf springs......This is also a F250 not a 150. And its a 4 wheel drive. From what I can find a 97 DID NOT have Twin I Beam design.That's a twin traction beam front suspension, so it's like a 4wd Twin I Beam design. The truck uses radius arms to locate the front beam axles. The radius arm bushings are shot. That's why I was asking about the 2wd/4wd thing previously. Only 2wd Twin I Beam and 4wd Twin Traction Beam trucks exhibit this behavior under driving and braking. Basically, the axle center is moving around front to back and not just up and down, the axle can travel in a circular or elliptical pattern too. This is not a ball joint issue.
Replace the rear radius arm bushings with a Moog K80007 problem solver heavy duty replacement bushing.
You'll also need to replace the bushings on the pivot end of the beam axle with a Moog K8620 heavy duty problem solver bushing.
Total parts cost from Rock Auto with shipping will be under $75. It's a hell of a job though and unless you have a good bit of heavy duty tools, including a 20-ton press, and solid mechanical knowledge, I'd suggest paying a pro to put them in.
Final thing, buddy up to a tire shop and let them put your wheels on the balancer. Make sure the tires and wheels are round, not just balanced. You need a fixed point of reference to watch them. If they are not true, sometimes you can pop the tires loose on the wheel, and turn them relative to the wheel to put the high side of the wheel opposite of the high side of the tire. Of course that usually requires balancing them again, but usually you'll find they take less weight when you take the time to do this, and the truck will ride better as well.
I went through all of this with a '93 F-150 SVT Lightning years ago, after the truck hit 100k it lost it's mind and was scary to drive. The above fixes, with the F-150 specific parts, resolved the issues completely.
Some of the light F-250's did have the drive axle mentioned. Your's is not a light duty version.I do NOT have a rear radius arm.... I have leaf springs......This is also a F250 not a 150. And its a 4 wheel drive. From what I can find a 97 DID NOT have Twin I Beam design.