Impropperly balanced tires usually only create a vibration.
There are numerous things that can cause the symptoms you mentioned, and it is probably not one single item. I suspect that none of the suspension components have been replaced in your car, and they are all probably past due. Start with the control arms and FCAB's, and the RTAB's. You probably want to change the shocks and struts as well, and all of the mounting hardware, especially the rear shock mounts. The last items to address would be the front hubs and bearings, and the rear bearings, as well as the sway bar bushings and the front and rear swar bar connectors. The front sway bar connectors really should be changed along with the front struts. All of this does not need to be done at once, but you should probably attack it in the order I listed, to get the most bang for your buck.