I am having this same problem!
I also removed the APP, removed battery, and re-installed, still no go.
I have spent all day trying different solutions to somehow save/record my voicemail messages why is this such an impossible task?
You can not record phone calls so you can't make a voice note of your messages and most "services" simmilar will only allow you to transfer messages received after set up and not ones currently saved on your verizon voicemail.
I may or may not be able to live without the messages my brother left for me when he was in Iraq, but the fact that something that should be fairly simple is nearly impossible to accomplish is beyond frustrating!