TCMgo

I drive a 1997 Jeep Cherokee Sport with 220,000 and running strong. I have been working on this mystery noise for three years! The noise has not gotten worse in all this time. I even posted on this forum some time back. Here is the noise and what I have done to date. A brief grinding noise upon accelerating after braking and coming to stop. After a few seconds the noise disappears. The harder the braking the louder the noise upon accelerating. If I roll to a soft stop (a near stop) there is no noise when accelerating.
I have replaced all u-joints and related drive line parts. I have inspected motor mounts and transmission mounts, replacing one suspicious transmission mount. I have replaced front rotors, calipers, pads, and brake hose (now braided stainless steel). Rebuilt the rear drum brakes, but have NOT replaced the brake hose, yet. Replaced all shocks, and rebuilt front suspension. I had thought that perhaps one of the brake hoses had collapsed causing fluid return problems. I don't think this is the problem.
By a process of elimination I have deduced (with my limited mechanical skills) it down to something either twisting or torquing upon stopping that grinds for bit until it straightens out upon accelerating. Someone said to look and see if any rear suspension parts like a stabilizer or anti-sway is missing. Since I don't know what should be there, I don't know the answer to that one. Are there after market devises to stabilize the differential? I am not sure what to look for. Please advise if you have had the same problem.