I probably shouldn't be writing this as I'm not a morning person, but I'm gonna click on the steps as I write.
As a troubleshooting step, assuming the new domain you registered appears in the Domains, Manage tab, note the Status. Does it say Active? Click Edit on the line where Service Type is Web for the actual domain.
Again, just as a temporary troubleshooting step, you could have the new domain point to the sub-domain content. (Last option on the Web Edit Screen. (A Mirrored Domain (a domain that mirrors another site's content / browser URL remains the same).) In the pull-down menu off the last option where you've placed a dot to select that option, make sure the sub-domain name is there and click Save Web Service.
Wait a few minutes and typing www.newdomainname.com (whatever it is) should show the exact same thing as oldsubdomainname.dreamhost.com
If all this works you know that the domain is active and that the old subdomain content is still there.
I come to the forum randomly but feel free to email me privately jim at pics-by-jim dot com. I'm not a dreamhost staffer obviously, so feel free to contact tech support also. But I'm here if I can help.