Normally, a "shield" conductor is used to cut down on interference to semiconductor devices. On a drum switch, where the load is a motor, that is not so much an issue. The shield wire should be grounded at only one end. In your case, at the drum switch would be the most effective. But space would be at a premium, so the motor end would probably be easier. As far as to what terminal, it should go to the metal frame of whatever it is attached to. Such as the frame of the motor. It should be connected at both ends
only when it is a "ground" wire. The frame bolted to the mount should take care of that issue.
I am assuming a lathe or some such. For the "stop-start" function on single or three phase, you will find that a six(6) conductor cable is necessary. Four for the start winding, two for the run winding. For a wound field DC motor, eight conductors will be better. For a permanent magnet DC motor, you could get by with four conductors. If you were using this strictly for directionals, with a seperate "start-stop" switch, four conductors would do. But remember that the motor has to reach zero speed to reverse on single phase.
Now, as to the 22 AWG cable
: 22 Ga wire is a bit small for motor leads. It would be better to use 16 AWG for 1/4 HP, or larger for serious motors. That would be your call though. A matter of what is available, in most cases. Just be ready for burned wires. I have a 1/2 HP motor with 12 AWG leads on my Craftsman 12X36. What I had when........
Bill Hudson