I can't speak to the functionality of the factory glow system, but no the plugs do not need to be energized with the engine running. Like Jim mentions, you can set up a switch to operate them on demand to help smooth out a rough start/idle. I wired mine so as long as the ignition is in the ON position I can use a switch to power the glow plugs.
Put a volt meter on your glow plugs with the engine running - it's unlikely, but possible for a relay to fail closed. I'd also check the wiring of the ignition switch. You can wire in a generic relay to use your "GLOW" ignition switch position, or use a momentary switch for the 'afterglow' functionality, or both.