New Primary Domain

Hi, I just changed my primary domain (hosted by bluehost) to something else other than my previous one. It was successfully changed/renamed. And then bluehost told me to move the public_html file to the new domain. However, they also told me that if I’m not comfortable doing it, I better ask someone else to do it. So I assume it is really difficult.
Then I went to settings > general, and changed the site url and wordpress url to the new domain name, and everything is done.
So now I’m just wondering, if bluehost support say it’s going to be difficult, and I’ve researched on the internet and everyone say it’s a difficult process, what do I have to know? What could happen to my blog, or what’s actually happening? Do I still need to move the files?
Also, when I search my blog on google, the blog still comes out as the old URL. How do I change that?
I need some suggestions on what to do, please. Thanks!

The blog I need help with is designisyay.com (formerly catgonemad.com)