The only time I get the saved voicemails first is when the 21 days has passed and the voicemail needs to be re-saved or deleted. I always quickly skipped to the end of the message with a double 3 command. I believe there is another method of skipping the message altogether, but I don't use it regularly enough to have it memorized.
I'm most definitely NOT a VZW employee. If a post answered your question, please mark it as the answer.