[Is it possible to change the URL of your existing gig?]

Gig permalinks stays the same as like your initial title when you created and published the gig.

I found it happened even before publishing. The first time you write a title in your draft the permalink is set right after saving or moving to the next step.

It happened to me but as @misscrystal says the title is still relevant. Actually I see it as a good thing since it’s a slightly different way to communicate what the gig is about (in case anyone bothers to look at the url)

If it happens to me again while drafting I would just recreate the gig before publishing.