When I plug a different headset into my computer while DDO is running, as soon as it is recognized by Windows it also immediately becomes available for use in-game as "Voice Playback Device" (and "Voice Capture Device", for that matter). However, it does not show up as in plain, old, "Sound Playback" unless I exit the game and re-start the client.

Please do whatever it is you are doing that correctly lets a new device be picked up for Voice Playback also let it be picked up for Sound Playback.