Never used Verizon's visual voicemail service, didnt feel like paying for it...
have you tried Google Voice? You can set it up so that it only gets used for voicemail, like i do.
It allows you to select specific messages to playback, and even attempts (amusingly) to translate them into text form.
Give that a shot, I've never had any issues with connectivity or late voicemail's (as far as i know)
EDIT: ITS FREE!