However, I note that: (a) This only happens when the engine is running and (b) When the high beams are turned on, the bulb warning light goes out.

For the above:
(a) The Bulb Failure Warning System relay is operative only with the engine running. The key in Position II with engine not running is the bulb test position, including the bulb failure light.
(b) The BFWS checks only the low beams, rear marker lights and three brake lights.

The BFWS relay trips when one of the pairs of lights has a current draw on one side unequal to that of the other side (rights side vs left side.)

Other posts have what you need - replacing in pairs, checking for good grounds, etc.

Good Luck,

Bob

:>)