This is just a stab in the dark... the DC offset may be steered to a high value capacitor that forms a time constant with a parallel resistor that builds up voltage to a transistor base to saturation to open/close the relay. If that capacitor is shorted or lost capacitance then the relay will see the DC offset condition.
Change all electrolytic caps in the area. And diodes, too. A diode could be rectifying voltage from the power supply secondary to that time constant transistor. That circuit is most likely separated from the center tap by another high value electrolytic cap and if that diode (or cap) is shorted it pulls the relay power circuit to ground, causing the same problem.
Change all electrolytic caps in the area. And diodes, too. A diode could be rectifying voltage from the power supply secondary to that time constant transistor. That circuit is most likely separated from the center tap by another high value electrolytic cap and if that diode (or cap) is shorted it pulls the relay power circuit to ground, causing the same problem.