Hello Stephen,

I would recommend any of the popular DAW's that can be used in windows. Here are a list of them:

• Pro Tools
• Reaper
• Fl Studio
• Steinberg Cubase
• Ableton Live

I have used Reaper and ProTools for this in the past. I find Reaper has the best deal, and is the easiest to use. 

I have the Adobe suite, including Audition. Is that a "DAW"?

Audition is a Digital Audio Workstation. 

So your desire is to attach a soundtrack to a video rather than to make a (screenshot) video of doing something in Finale.

I should clarify. I want to capture the video (and audio sometimes) when playing music in Finale. There is a moving vertical bar and the music scrolls. Sometimes I'll play the music and record it, and I'll use Adobe products to edit in my audio. I think the DAW solutions will only capture the audio?

To do what you want, you would need a different program from those suggested so far. Unfortunately I am not familiar with this aspect of A/V creation.