It sounds like you have either a Deluxe or higher hosting plan.
In that case, you can add multiple domains to the account, which you may have already done. http://go.me/kl
Each domain designates a folder as its own root. When you install WordPress into each of those domains' folders, you'll still just specify the domain itself and leave the directory field empty. Once you've installed it, you should be able to restore the database for the installation and it should appear as a standalone site.
If I'm understanding your question correctly, you may already have this set up properly and the links are broken when you change your site address. Resetting the permalinks (disable/re-enable) could solve the issue with the broken links.
If you have a ticket with support that I can review, please feel free to post it here.
^Cj
