My car has 160,000 miles. Recently, I've noticed that buzzing and grinding sounds started coming from the Police Interceptor at RPMs around 3000. Also, when I am driving on the highway, there is a significant shudder during acceleration. This occurs only when the transmission is not in overdrive. The reason for that might be that I manually disabled the overdrive. It is either that or because it downshifts out of overdrive as I give enough gas. The transmission was rebuilt. Besides that, I changed oil cooler, condenser and transmission cooler because the transmission fluid was leaking. I cannot hear the sound when the car is parked or in neutral. This problem occurred suddenly. I did hear some quiet buzzing during driving last month, but this grinding sound occurred very recently and it could be heard occasionally.
The noise is coming from the internal transmission bearing or overriding clutches. Not all of these parts may have been replaced when the transmission was overhauled.
I can only recommend having a qualified transmission mechanic come and listen to the grinding noises and other problems you are having. Just the fact that you disabled the overdrive for the transmission could have caused undue stress to the transmission and should be fixed also.