Unless you plan on letting your old domain expire, the only thing you really need to do is set up your old domain to permanently redirect to your new one, and ensuring the redirect keeps any URI/query string parameters in place.
If you'd rather have references to your old domain replaced completely, you would need to run an SQL query for this as well as have your search index rebuilt.
This is something that's outside of IPS' standard support scope as they told you. I'd be happy to do this for you as a one-time paid service if you'd like to PM me here. It should be no more than 2-3 billable hours of work to complete. If you prefer to do it yourself, I could find an example SQL query for you to follow, just bear in mind you just have to be extremely careful and always make backups when dealing with anything SQL related.