Make sure your reverse lights aren't on a relay. previous owners who add switches, may have added relays. And cut wires.
The connector from the main engine bay harness, to the tranny extension is behind the large cowl fresh air tube. Make sure it is intact/plugged in.
(I think it was a four wire plug, and the orange was for
Transfer Case switch.)
Anyhow.
The Three pin round plug.
Red should be hot w/ignition on.
White w/ trace is backup lamps, Just like in rear.
Black is ground to solenoid.
It connects to the starter solenoid bottom post. I have a standard tranny solenoid, without post, because sometimes you need to get the motor running after a stall-
Quickly.
The switch grounds out through the body of the tranny switch, to provide a ground for the solenoid. Without a ground, the solenoid doesn't work.
You
can wire a secondary switch for backup lamps, anywhere on the White w/trace wire. I have mine done up this way. I'd use a relay, and separate fuse doing it his way, though.
Also Your shifter linkage may be shot. My brand new switch wouldn't work even remotely correct until I rebuilt my shifter linkage. The linkage supports the shift arm, which
must have no wobble in it's travel.
Any lateral movement = little / no contact with switch.
Mine was a Real PITA to rebuild, Yours is probably more accessible. Kits are available, but the parts need some tweaking to work.
I needed two kits to finish one complete assembly.
I run a Column shift on mine, though. and 727 tranny has same shift arm as 999.