[joe231]

Markeb01 is correct, it is the ground side that does the switching.
ground side switching is very common in many years of automobiles.....

there is a separate contact in the headlight switch that completes the dome light circuit. It is a white wire. unplug your dome light connector and use an Ohm meter to check the resistance of the white wire to ground with the headlight switch in the dome light "on" position (while you have it unplugged check the condition of the contacts in the switch, look for corrosion). It should be close to zero ohms.
Is the switch mounted in the dash? I believe the switch gets the ground from the mounting in the dash.
If you don't have a good ground at the plug, then check at the switch itself, if you have a good ground n the white wire at the switch and not at the plug you probably have a bad wire between the 2.