Which phone model do you have? Are you using Basic non-visual voicemail or a Visual voicemail?
I'm most definitely NOT a VZW employee. If a post answered your question, please mark it as the answer.
Just to confirm,you have basic voicemail (dial *86, enter your password, listen to messages)? While listening to a message, you press the number 7 - you SHOULD hear "Message erased, next voice message". What do you hear?
You SHOULD be able to go through each of the messages, listen to a word or two, then press 7 to erase, and go on to the next, and eventually clear out your voice mailbox. It can hold about 20 messages, unless someone has left you an extremely long message.