I don't think she would ever come back to take Cody, and if she did, it would only be for a little while because she wouldn't be able to cope with actually having to take care of him herself, having no one (Zach) to dump him on when she wanted to go out drinking with her friends without having to spare a thought for Cody.

Also (something I'm planning on working into a fic), Zach would need legal guardianship (at least temporarily) of Cody to be able to do things like enroll him in school, take him to the doctor, etc, and I'm guessing that Jeanne would sign off with few qualms, one more problem taken care of.