Publishing website

Hi all,
I’m nearly finished my new website but there is one thing that is annoying me a lot which i cant seem to fix no matter what i try or look up on google.
Basically i already have another website called ballyadamsparish.com
I want to replace this website, with my new one (the WordPress one).
Every time i click publish, it publishes the website, so then i go to ballyadamsparish.com and its still the old one (the blue one)
Also, yes i set my site address URL to ballyadamsparish.com, however when i did this, every time you went to the website, it would randomly choose which website to show (the blue one or the new one). Anyone got any idea how i can delete my old website or change the URL? It wasn’t made with WordPress so that’s why I’m unfamiliar with how to change it. Thanks guys

The page I need help with: [log in to see the link]