The interior lights and others controlled by turn the headlight dimmer all of the way is connected to a switch in the headlight switch. There are several different and separate switches in the headlight switch.
The dash light dimmer operated by rotating the headlight switch, but not all the way to activate interior lights, is a variable resistor. That power comes from the parking light circuit, but not the headlight circuit. So the park or running lights must be on first. Power comes from that circuit to a usually 5 amp fuse and returns to the headlight switch resistor to feed the variable dash lights.
The park/running lights work so the power feed is good. Check the separate 5 amp fuse (maybe 4, maybe 7.5) If that is good, back to the resistor. If the power to the lights is good but the problem still exists, be sure to check the ground wire attached to the gauges/speedo/tach units. They all run together into one ground point. If it becomes weak it will affect all of those lights.
The dash light wire is light grey, the ground is black.