I think I’d start something new, but salvage as many ideas and characters of my own that I could from the old story. I’ve experienced similar situations and it’s always awkward. But I don’t think I could ever write about someone else’s character, because it just would feel like I owned it. If you do decide to continue on with the current project, be sure to get her permission (written, perhaps?) to use her characters and story ideas, otherwise she could cause you problems if you decide to publish.
And never give up. I feel like giving up all the time, but so far I’ve somehow kept plugging along. If writing means enough to you, you should (and will) keep plugging along as well.