Assuming it does affect the dome light the relay could be internally shorted or contacts welded together and that could provide the short to ground keeping the dome light on.
The diagram I posted of a 74 valiant (post # 17) shows a pink wire going to the dome light. It comes from the fuse block. It provides the 12V to the bulb, the yellow wires all provide the ground when a switch plunger comes out ( ie the door is opened) or when the dimmer is rotated clockwise till it clicks, the diagram does not show the function of the "Accessory Conn & Time Delay"
