Websites are not functioning after migration from Plesk or other product with external or centralized SQL server to Plesk with local or new external SQL server.
Plesk does not make any changes in the website content and the connection strings are not changed to new location of SQL servers.
You can search for connection strings in web site content (web.config or CMS config.php e.t.c) and change them manually.
Also you can find all connection strings using the following commands:
findstr /S /I MySQL_SERVER_NAME.TLD *
findstr /S /I MySQL_SERVER_IP_ADDRESS *
findstr /S /I MSSQL_SERVER_MAME.TLD *
findstr /S /I MSSQL_SERVER_IP_ADDRESS *