So of course when I awoke I tried to look up this musical, and it turns out it doesn't exist.
The musical is called "Annabel Lee", apparently based on the poem by Edgar Allan Poe. But the Poe poem isn't the opening number. The opening number, which is one of those small-town-and-market-intro settings, is clearly inspired by the opening number "Belle" from "Beauty and the Beast" (lyrics by Howard Ashman, music by Alan Menken). The actual tune from the opening number, though, turns out to be the music written by Harold Fraser-Simson for A.A. Milne's poem "Cottleston Pie" (as sung by Rowlf the Dog from episode 7 of the Muppet Show). Of course the refrain is "Annabel Lee" instead of "Cottleston Pie", and unfortunately I don't remember the rest of the lyrics from my dream.
Usually when I have a strange dream like this there are elements from things that I did or watched or heard recently. But I swear this one came out of nowhere.