How do we change the URL of a working GitLab install?

You did everything correctly!

You might also change the email configuration, depending on if the email server is also the same server. The email configuration is in gitlab.yml for the mails sent by GitLab and also the admin-email.

GitLab Omnibus

For an Omnibus install, it is a little different.

The correct place in an Omnibus install is:

/etc/gitlab/gitlab.rb
    external_url 'http://gitlab.example.com'

Finally, you'll need to execute sudo gitlab-ctl reconfigure and sudo gitlab-ctl restart so the changes apply.

---

I was making changes in the wrong places and they were getting blown away.

The incorrect paths are:

/opt/gitlab/embedded/service/gitlab-rails/config/gitlab.yml
/var/opt/gitlab/.gitconfig
/var/opt/gitlab/nginx/conf/gitlab-http.conf

Pay attention to those warnings that read:

# This file is managed by gitlab-ctl. Manual changes will be
# erased! To change the contents below, edit /etc/gitlab/gitlab.rb
# and run `sudo gitlab-ctl reconfigure`.