Hello @YuriiP ,
It seems that datastore is being pointed/referenced using IP address in comparison to ArcGIS Portal and ArcGIS Server (which are being referenced through FQDN).
This can be confirmed by navigating to ArcGIS Server Admin > data > items > enterpriseDatabases > Select "/enterpriseDatabases/AGSDataStore_ds_xxxxxx" under "Child Items" > "machines" beside "Resource". If it shows IP address ArcGIS Server is utilizing IP address to connect to ArcGIS DataStore. (This needs to be performed on current environment/source)
As per my understanding the IP address of the current machine is surely different from the new machine.
Furthermore, I have encountered the problem in the past which I was able to workaround using few machines changes but also caused damage. Hence, I would not recommend any of those as they can cause an impact on current environment.
I would suggest switch the WebGISDr log levels find more details what is causing the process to hang utilizing the "Valid levels DEBUG, INFO, WARN, and ERROR"
Additionally, reaching out to ESRI Support services could also be helpful.
Hope it helps!
Archit