You didn't mention changing the Name Servers wherever you have registered your domain. They must be changed from pointing at your old host, to pointing at your new host.
If you've done that, it can take up to 72 hours for everyone in the world to see that change in Name Servers, so you need your site to work in both places during that 72 hours. The changes make their way gradually to the tens of thousands of Domain Name Servers (DNS) across the planet.
To make things more complicated, DNS information is also stored on your computer, and pages can be cached in your browser, so you can not always be sure that you are viewing current information.