you have the old db-server up and running, use the following command in PowerShell:
Get-SPSite -ContentDatabase <SourceContentDb> | Move-SPSite -DestinationDatabase <DestinationContentDb>
This command moves all site collections from the source content database to the destination content database.
Reference: Move site collections between databases (SharePoint Server 2010)
depending on where in the process you are in moving, My best guess is that you have to mount the database(s) using the following PowerShell command:
Mount-SPContentDatabase -Name contentdb -DatabaseServer dbserver -WebApplication webappname
When that is done, continue with a test and solve any errors listed by the cmdlet
Test-SPContentDatabase -Identity contentdb
Reference: Managing SharePoint Content Databases with PowerShell
To move all databases without the use of SQL Server alias, I find it faster (and easier) to set up a new farm on a different servers and do a full backup and restore in the new environment using PowerShell. Don't forget to use the -overwrite flag.