I got a new cell phone recently. As the world stood in line to pick up the latest iPhone 8, I graduated to the 6S. I pride myself on that actually. As with all new phones in the 21st century, the voicemail must be set up anew. I used the default message for the first few days, but finally settled down to record a more personal greeting.
After putting phone in airplane mode, deleting message, I had to turn phone off and re-boot. It seems to have worked!

Keep callers on the phone with messages on hold. Make announcements while they’re waiting or being transferred.
Great question! Although I don’t know if there is a specific function for this on your particular phone, the best way to save the message is the simplest. Turn up the sound as high as you can and record it onto a digital recorder (like you would use for a lecture), a cassette recorder or your cell phone if you have a voice recorder on it. Then save it as an MP3 on to your computer for safe keeping. You can also Google your vtech model number to see if you can find an owner’s manual. They might have a way to save a recording in there. Probably not, but it’s worth a shot.