You need a new multi-function switch (P/N 5269377 SWITCH, Multifunction)
Twelve volts is always applied to one end of the bulb filaments, Your headlight switch is doing its job of providing ground for the headlamp circuit. The multi-function switch "decides" which of the bulb filaments is hooked to that ground. The multi-function switch is not doing its job.
Your headlight switch is okay.
Edit:
Before you buy a switch, see if adjusting the steering wheel tilt through its extreme positions has any effect on the "no lights". Some on this board have reported loose connectors/broken wires at the multi-function switch.
Good luck.