Just wondering what type and rev of Partner system is being used? (Partner ACS, Partner II).
If the problem is affecting all extensions and shutting off the MIC/HFAI function when checking via speakerphone, or via the handset doesn't work and the factory prompts sound fine, you could try to reset and/or reseat the voicemail module. If that doesn't work, usually you then end up replacing either the voicemail module or the Partner processor as either device could negitively impact voicemail message playback.