I recently transferred my domain name to GoDaddy and sorted out the DNS stuff to hook it up to my new site which is hosted by somebody else but when I try to look at the site myself I get the message 'hazelbrownart.com's server DNS address could not be found.' although when anybody else searches for hazelbrownart.com they can access it.
I cleared my cache as suggested and tried a different browser and still nothing. I got my housemates who are using the same wifi to check it and they can't access it either so I believe its something to do with my internet but I'm unsure of how to fix that?
Try clearing the DNS cache on your PC and browser. See this article for instructions for various operating systems and browsers: https://support.opendns.com/hc/en-us/articles/227988627-How-to-Clear-the-DNS-Cache-on-Computers-and-...
Already cleared from my browser but I've just tried clearing on my PC too and that hasn't made a difference.
How long has it been since you changed the nameservers? I had to change the DNS in my router because my ISP's DNS took so long to propagate. I now use Google's public DNS servers.