I think maybe you are not really understanding what is happening with the domain names. a "firstname.lastname@example.org" domain name is an entirely different domain that one that your register, and they each have their own directory under your user on the Dreamhost server.
What you really need to do is not so much a matter of "linking" the site to one domain or the other as it is setting up the files in the appropriate directory so they they are served when DNS looks up the domain named that is entered.
Assuming that both the dreamhosters.com and the new domain name are under the same user, and depending upon what you have installed there, the easiest way to to do what you want is to just rename the directory "whatever.dreamhosters.com" to "younewdomain.tld" - then, whatever is shown in that directory will be displayed when a visitor browses to younewdomain.tld"
That said, there are some caveats here: If your content is driven by scripts (WordPress, Joomla, etc. ) there are other steps you must take to modify your configuration files so that they reflect the new domain name after the re-name of the directories, or they will not work properly.
An additional approach to dealing with this is to simply re-install your applications/files in the new directory created for the "yourdomainname.tld".