Excellent. That is the best circumstance to be in here, beacuse if all else fails you can go elsewhere.
You can create a new dreamhost account without specifying a domain. I haven't looked at the sign-up process lately, but there is a way to say "I'll add one later." I would do that, because then you can create tickets the normal way and chat with support instead of using the contact form. (I think the tickets that get created by the contact form go someplace ultra low priority).
Be certain that support understands that you (not the owner of the other dreamhost account) owns the domain and that is registered at godaddy. They should notice that themselves, but point it out before they make a bad assumption or guess and send you the wrong answer. They may ask you to prove you're the domain owner, they have a proceedure for that.