Are you using visual voicemail or the basic non-visual voicemail?
I'm most definitely NOT a VZW employee. If a post answered your question, please mark it as the answer.
Ann154 asked a great question. Can you confirm the type of voicemail are you using? Or, is it Visual Voice Mail or Basic Voicemail with dialing *86? To change the greeting, you will simple dial *86 and select the 'Greetings" option to change the recording for standard Basic Voicemail.
To set up voicemail:
For current Voicemail Instructions, see Voicemail Instructions.