Ok, I figured it out - turns out the DNS WAS set incorrectly after all! I was able to narrow down the issue when I started creating a bunch of dummy sites and playing with my local computer's hostfile and saw that everything seemed to be routing just fine. So, clearly IIS was working as expected. So, I then called GoDaddy repeatedly until I found someone who agreed to help me with a Google domain. Apparently, from what I could understand, I set the IP address in the wrong place which only made the domain forward as the IP address, so there was no hostname for IIS to match it to, which is why it pulled up the default website (no bindings) with no problems, and ONLY the default website! He directed me to another screen (the DNS Zone File tab) where I could set it properly. Now, everything goes to the expected sites!
Hopefully this will help another newbie out one day when they have to set their own DNS (gasp!)