Changing SQL Server service account passwords enterprise-wide
Option 1

I would probably take advantage of using a Central Management Server to run the password change script across all the different servers at once. This should provide an easy method for verifying the output from each server while keeping it easy to manage by executing once for all target servers.

Option 2

If I was doing this in a tighter change management system, then I might consider scripting it as a batch file and having a sqlcmd execution output a separate log file for each server. This would verify the changes and keep record of them more reliably than doing it in SQL Management Studio.

I don't think there is necessarily a "right" method here, just depends on what meets your requirements best.