This really should be a comment, but haven't got the reputation yet.
There's no way that you can have multiple domains running on the same host if the hosting company doesn't support adding extra domains to the host. The underlying webserver needs to know where the files from a domain is located.
Depending on the host you can sometimes create subfolders within the main folder that behaves as a seperate domain and other times you'll get another folder withing your top-level folder.
Besides that there is several systems that should allow you to do that. On top of my head I think that the newest wordpress version can do that, Silverstripe cms and Jomla should also be able to do that.
I'm aware of your limitations of the host. The CMS that I listed do support this but not all of them right out of the box, sometimes you need to make small hacks to make it work.
For wordpress you need v3 and see:
For Joomla see:
For Silverstripe there is a module called subsites which allows this documentation can be found here:
I have used this in the past, but it may not work with the latest version of the Silverstripe
I think Drupal has this feature as well see this
Hope this helps to give some alternative