Can a GCP organisation change its domain?

You can change the organization's name but since you already have a new domain added then the only thing that's left is to migrate everything to the new one and make it a primary domain.

The actual "switching" is easy but there's some work to be done beforehand. Otherwise you may end up with

• bounced emails
• not being able to log in to your GCP account
• some 3rd party sites (ebay for example) use Google SSO to sign in - they may break after the change.

Doublecheck everything before you actually change your primary domain !

This has been discussed some time ago on SO here which may explain why it's not that easy to find how to do this.

Additionally it's worth watching this YT short explanation how to change primary domain and what are the prerequisites and effects.