Many thanks to those of you who responded with suggestions. I tried the super-search as per the first suggestion and had some luck this time. I was able to find a match to at least three. There were two that I couldn't find a match to, but I feel reasonably certain that she wasn't a composer, but I do know that my second great grandmother did enjoy the music of her day.
The best part was being able to hear the simple tune of the songs directly from my computer! What a neat twist on preserving something of my past. I have a copy of the one and only photograph that survives of grandmother Alice. I got it out and showed it to my daughters and then played the song on the computer. "This is your third great grandmother and this is a song she used to listen to," I told them.
The trouble I had when I first tried was that the titles Alice had written down were different than what was in the database and some words were a little different but that can be attributed to regional differences I guess. Would there be any benefit in having these different versions deposited in some repository somewhere?