Hours and hours of frustration on the phone with Roxio led me nowhere. They claimed to be completely unaware of the issue, but when I dug around the internet I found it was a known issue that began with Mojave.
I did solve the problem, but am not exactly sure how My research came across some references to "soundflower," which I had on my older Mac, so I got the feeling that was part of the puzzle. I found a place to download and install it for free (someone named Matt Ingalls took it over so google him and soundflower and you should find the link, as well as some articles on installation). I'm kind of puzzled because I don't have it selected as my input device, but I can now use Live Screen Capture for video with sound.
Do some research before downloading it, though, 'cause I'm not sure what OS you're running (I'm still on Mojave), and don't know if it'll cause any negative issues for you.