Please follow these steps.
1. Close all instances of Citrix Studio. Any configuration changes (even through PowerShell) must be stopped while following the steps (You can power down DDCs to be extra cautious).
Take VM snapshot or take backup of all Delivery Controllers.
2. Take full backup of Site, Monitor and Logging databases.
3. Restore Site, Monitor and Logging databases from backup to the new SQL server.
4. Add databases to Cluster, Always-on or Mirror (if applicable).
5. Create logons for the Delivery Controllers on new SQL server(s) and permission should like this.
-- In this example Databases are Citrix1912-LTSRSite, Citrix1912-LTSRLogging and Citrix1912-LTSRMonitoring







6. Power on all Delivery Controllers and note down DB connection Strings for Broker service and Logging and Monitor datastore.
- Open Administrator PowerShell on one Delivery Controller and execute following commands
-- Add-PSSnapin Citrix*
-- Get-BrokerDBConnection
-- Get-LogDataStore
-- Get-MonitorDataStore



7. Make sure there is no connectivity issue from Delivery Controllers to new SQL Server / Cluster / Always-on Listener.
-- You can create a file with .udl extension, open it on the Delivery Controller and test connection as follows.



8. Prepare DB Connection strings to set though PowerShell. If you are using specific SQL instance and non-default SQL port connection string will be like this.
$Site = "Server=<NewSQLServer\InstanceName,CustomPortNumber>;Initial Catalog=<Site Database name>;Integrated Security=True"
$Logging = "Server=<NewSQLServer\InstanceName,CustomPortNumber>;Initial Catalog=<Logging database name>;Integrated Security=True"
$Monitor = "Server=<NewSQLServer\InstanceName,CustomPortNumber>;Initial Catalog=<Monitor database name>;Integrated Security=True"

For default instance and port connection strings will be like this.
$Site = "Server=<NewSQLServer>;Initial Catalog=<Site Database name>;Integrated Security=True"
$Logging = "Server=<NewSQLServer>;Initial Catalog=<Logging database name>;Integrated Security=True"
$Monitor = "Server=<NewSQLServer>;Initial Catalog=<Monitor database name>;Integrated Security=True"

Example:
$Site = "Server=MYSQLSERVER.DOMAIN.LOCAL;Initial Catalog=MySiteDatabase;Integrated Security=True"

9. Download the script attached and update line 85 (Step 10), 103 (Step 11), 108 (Step 12) and execute sections as instructed in the script.
10. Restart Delivery Controllers, open Studio.
11. Check VDA registration and launch.

Please note that if the change is linked to a product update, there was a VUS service introduced as a FMA service operating on the Delivery Controller since release 2411.
Upgrading the Site from version 2402 to 2507 is taken as an example. When some of the Delivery Controllers in the Site had been upgraded to 2507 and the Site upgrade was performed, the database schema was upgraded to the corresponding version, and all FMA services on those Delivery Controllers were configured successfully, including the VUS service.
However, the remaining older version of Delivery Controllers had not yet installed the VUS service because they had not been upgraded to the 2507 release yet.
Therefore, after upgrading the remaining Delivery Controllers, it is required to close and reopen Site Manager. Site Manager may then prompt the Citrix administrator to perform an additional Site upgrade to register all the Delivery Controller’s services to the site.
This is also documented here: https://docs.citrix.com/en-us/citrix-virtual-apps-desktops/upgrade-migrate/upgrade#step-9-upgrade-remaining-delivery-controllers