Every writer I've ever spoken to feels fraudulent in some way or other.
For me it's more fun to find an unexpected moment for a character to sing when you don't expect them to.
The worst thing you can do is censor yourself as the pencil hits the paper. You must not edit until you get it all on paper. If you can put everything down, stream-of-consciousness, you'll do yourself a service.
And this, from "Send in the Clowns"...
Just when I'd stopped
The one that I wanted was yours,
Making my entrance again
With my usual flair,
Sure of my lines,
No one is there.