If you bead the control cable, use mix 31 (DX Engineering carries them) and make at lease two passes through the cable on both ends.
You should be connecting the ground at the battery. Through the body grounding can cause a ground loop, and they're nearly impossible to find. Further, the voltage drop could be enough to cause the problem as well.
One more comment. The head of the 706 is isolated from ground. If by chance the shell of the adaptor is grounded, and it touches the grounded shell of the pickup in the cassette player, it could exacerbate the problem.