Can anyone help me? I registered a domain already in use by someone else

I’m so irritated! I’m brand new to all of this and barely even know the basics. I’m using Dreamhost for basically everything with Wordpress to… host? So, everything was working fine, I was editing my blog under a domain/URL that I didn’t really like. Dreamsofdrea.com Once I was done editing, I decided to change it to something catchier which was bookedallweekend.com since my blog would be about books, reviews, and movies. So, I followed the guide on dreamhost and registered the new domain name and basically everything went to crap.

Dreamhost ALLOWED me to use bookedallweekend and it said it was available! So, I click on the website and it turns out, it is being used by someone else.
When I contacted customer service, they told me that my DNS/name server are pointing to godaddy (I’ve never heard of it and don’t know what that means), and I just need to change it to Dreamhost.

I’m THINKING that the OTHER girl’s site is the one pointing to Godaddy and any changes I make would alter HER site, not mine?
Unless, our domains are automatically purchased through Godaddy and I’m just confused? Can 2 people register with the same domain? Why did it even show available?

Can anyone help me with this problem? I don’t know if I should simply change my domain name, AGAIN, or what?