I'm assuming that when you say "domain only" you mean that you only have the domain registered at GoDaddy (not hosted there) - If that is not the case, the following won't be exactly right - just post back and clarify, ok?
You will probably be pleased to find out just how easy this is to do! Because Dreamhost allows you to host unlimited domains on your account (think or your "account" as being your user, not your domain) irrespective of where the domain is registered, you can host "run.com" right along-side "sneeze.com" with no additional charges.
1) From the Control Panel, go to Manage Domains and select "Add a Domain/Sub-Domain" at the bottom of the page.
2) Add "run.com" as a "Fully Hosted" domain.
3)Go to GoDaddy and, using their tools, change your DNS (Domain Name Server) information associated with "run.com" to the Dreamhost name servers (ns1.dreamhost.com. ns2.dreamhost.com, and ns3.dreamhost.com)
You are done! Your "run.com" domain will still be registered at GoDaddy.com, but will be hosted at Dreamhost. They will both be in your same Dreamhost "account", but when you ftp into Dreamhost to manage you site(s), you will see directories for both sneeze.com and run.com. Just build each site in the indicated directory.
Remember that the DNS change for run.com will take a while (anywhere from minutes to days and it varies from user to user and computer to computer) to be distributed across the internet before you can visit run.com with a web browser, but you can still upload files to run.com via ftp while you wait for that to take place.
If you want to "test" your run.com site before the DNS changes, you can do so using the instructions provided in the Dreamhost Wiki at:
You can also can pop on over to the Control Panel-->Goodies-->"One-Click" installs and set-up WordPress on your new run.com, and be "good to go!" Good Luck, and welcome to Dreamhost!