This is just spitballin' here, but what if your shocks have a dead spot in them? I have seen and OWNED some shocks that worked fine in most situations, but when I removed them, they had some free play in them.
Also, Is it possible that you have NO pinion snubber and the axle radically ROTATES when the tires get a good bite? Basically, the axle yoke where the driveshaft connects tries to go up and down rapidly? If that is the case, maybe a fabbed up pinion snubber would help.