It’s true that SharePoint recognizes the domain that the computer is joined to, but I’m not sure that SharePoint will resolve the domain change. And more importantly, user ZZZ\user1 is not the same user as XYZ\user1. This can be a real issue if you have a lot of content based on the previous AD. It’s possible to overcome, but it’s a lot of work. Sometimes it cannot even be performed properly I know from experience (at least in a 2010 environment).
It is fully supported in SharePoint 2013 to use ADFS. You should install and configure the ADFS with the web application(s) as a relying party. You also need to configure your SharePoint environment to trust ADFS as an identity provider. You can have several different identity providers, such as AD, ADFS or even Microsoft Account (Live-ID). You also need to use claims authentications (default from Central Administration) and ADFS as the identity provider.
It’s a delicate task to implement, but it really works even if it’s a tough task the first couple of times you do it. See more at Configure SAML-based claims authentication with AD FS in SharePoint 2013