Yes, The Client that has already been registered and has a domain change will have to be re-registered as the domain name is required especially while scheduling tasks and using the Auto-Login feature.
You can verify the correct Domain Name in the Machine Information section of Control Room for that particular Client.
The Client that has had a domain name changed can do any of the following (in the sequence mentioned):
1. Change the Application Path in Tools > Options > Advanced Settings. This will ensure that all the data that was available in the earlier path is available to the Client in the current domain.
2. Create a New Client. However, it is recommended that you use this option as a last resort.