Is the replacement light LED? Is so, try reversing the wires. Polarity must be right for LEDs. Otherwise, you may have too much resistance in the old wires. They show continuity, but there is too much power loss for the bulb to light. In this case, you’ll need to pull new wires.All this assumes that you’ve checked the easy stuff first, like good bulb in light, etc.
OK, so I understand.When you have the switch in the "Nav Lights" position both bow (red/green) and stern (white) are lit but when you throw the switch to "Anchor" the stern white does not stay lit (bow is not lit either) correct?