Shield ground to metal body or signal ground?
I'll bet the additional 'grounding wire' is actually tied to the shield (ohm it out), if it's not, it is likely that its a signal ground. If it is a shield ground, then treat it like a shield, ideally the cables should be shielded all the way to the equipment. 

If its a signal ground, it depends on what kind of signal it is. If it's differential (like ethernet, or RS485) then it needs to go down to the digital driver/receiver. 

If your running the cable through a slip ring, then run the shield on an additional conductor if there is one available on the slip ring, then to the shield. The problem with running cables through slip rings is it allows for the signal to pick up noise because there is no shield (shield protect against electric fields). However, this is the best you can do.