Welcome to the site from California.
Dome lamp gets fused battery hot (orange wire) all the time. The headlamp switch generally completes the dome lamp circuit to ground via the small gauge white wire. Check out the wiring diagrams in the technical section of this website:
http://forum.73-87chevytrucks.com/smforum/index.php/topic,11766.0.html...I've check the switch with a circuit tester and all the wires are hot....
If (as you stated) the white wire probes hot (measures ~12 volts), the circuit is complete from battery, thru the fuse and bulb, all the way back to the headlamp switch. Make sure the switch frame is properly grounded. If everything else checks out, the hdlmp switch may have gone on vacation.
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Again, glad to have you on board!