Let me recap what I think you have said. The problem is the shaft from the IP on which the TPS/FIPL mounts. Correct?
Then you've said two things which, in my mind, contradict each other. You've said that the shaft has snapped, and you've said that the shaft is loose. Which of those is what has happened?
The TPS/FIPL mounts on that shaft, and the TPS/FIPL controls the E4OD transmission and it's shift points. If the shaft has indeed snapped the TPS/FIPL loses all control over the transmission. I have no idea what happens in that case. I don't know if it is driveable, or if it will harm itself when driven. If the shaft is simply loose it will retain some semblance of control over the transmission. Again, I don't know what effect that will have, but I think nothing good. It may still be driveable to get you home, though the shift point may be rather strange. I certainly wouldn't want to drive it more than necessary to get you home.