Long shot here... . does your truck have a posi or no spin rear end. Back in the day when I was young and foolish, I had a Mustang with a posi rear end. After one too many stoplight races, the car started doing what you descibe, what I also would call squirming when taking off from a stop. I killed the posi clutches evidently as it left me dead at a light finally, beaten by a Camaro. I was disgraced... ..... :{
Sam
You could be on to something. If the rear axle has a factory Limited Slip Differential (LSD), and if the gear oil has not been changed in a while, it could be getting a stick-slip condition. The stick-slip condition is best felt in an old Jeep Quadra-Trac transfer case, but it can also show up--in my experience, I am not a mechanic--in a fairly high mile LSD. This happened to me with a 1976 Jeep Cherokee 4 speed with conventional transfer case, but with LSD. It was a weird feeling.
1. If the differential oil has not been changed in 50,000 or more miles, change them both now. Regardless of the brand you put in, buy a small bottle of LSD additive from the dealer or someone reputable and add it. If your truck has an oversize differential cover, add two. If the bottle of 80W-90 or 85W-140 says that it works on LSDs, add it anyway. (Do not ask how I know this. ) That might solve the problem.
2. If it does not solve the problem, try this. You may need to do this anyway to get the new oil into the LSD clutch packs. (A LSD is very weird inside compared to a regular differential. ) Find an empty parking lot. Paved is best, but dirt or gravel will do. Drive in fairly tight figure eights for about ten minutes. That will get the oil circulating. And it should solve the problem, if that is actually what the problem was.
3. If you bought the truck used, or if you otherwise know that it was used as a grocery-getter, try step 2 first.
4. If it is an automatic, you may be getting internal slippage that starts to go away as it warms up.
5. Double check the spring shackles again, anyway. While doing so, check the rubber "eyes" that the bolts go through, front and rear. The shackles and the front track bar (especially) can look OK, but the rubber can be shot and in need of replacement. They can make the truck feel squirmy, but it should squirm all the time.
6. With the parking brake firmly set, and the auto transmission in Park (if it's automatic), with the engine on and with you lying on the ground, have a trusted friend turn the steering wheel about 45 degrees each way while you watch the front end parts. If the diagonal track bar acts "funny" where it attached to the frame, it is shot. Replace it. If the right front tire, and only the right front tire, seems to be wearing on the outside, it is an indication that the track bar is going south.
Apart from that, I am clueless.