Could use some help on what should be a simple LED wiring scenario
If headlamps are tied to 12V with low side switches, the LEDs must also be tied high with common anode instead.  The lamp resistance is powering the LEDs.   Nice job on MGB. I had a '67 B when they had a positive ground system.

Note in your schematic, headlights are tied to +12 and switched to Gnd via relays. Thus when connected to LEDs with Common Cathode (CC) tied to ground via a resistor , they will stay on due lamp resistance to V+.

Therefore replace CC LED to gnd with CA LED to 12V and pull down LEDs with lamp relays to make this work.

Edit
I just realized, that  I misunderstood the modifications added two more Relays rather than being wired to existing Relay sockets. With this in mind, I suspect there is a lot of EMI on the wires which is sufficient to power the LEDs. This noise can be suppressed with >=0.1uF capacitor across each LED.  Are these HID lamps? That may do it.