NadiaC

It may take a few days or more to settle down. When testing, try using an incognito window to remove any caching on your computer.

You should delete the two records in red from the third-party domain hosting.
My guess as to why the old website is loading is a cache issue, which should be resolved when the domain has fully finished changing to the new website.

Thank you for your answer! I have fixed the DNS error by deleting those entries. Fingers crossed I haven't messed up anything else 😉 
I'm still having issues with the old site appearing on safari when I use the sitename without the www. but I'm hoping it's indeed a cache problem. I'll give it another try tomorrow.