You could also edit your hosts file to override DNS settings, like the same article suggests:
Instead of doing a DNS lookup, your computer also has a so-called hosts file. This file can point domain names to IP addresses on your own computer, and will be checked before any DNS lookup is done. So you can change the hosts file of your computer to point your domain name to your new IP address, before the DNS cache has cleared (or even before any DNS changes have been made).
HowToGeek has a great article on how to make changes to your computer’s hosts file here. You can find the website’s IP address to use by following this guide.
And then clear your cookies, browser cache and computer cache again.