Seamless Transfer of Web Site

From myhosting.com Wiki

Once your payment is received and your account is activated, you will be able to upload your files and create your email accounts using the IP Address of your account. Once you have completed the uploading of your site, you can then proceed with getting your domain name switched over to our DNS servers.

When you first change the Name Servers, email will go to both the new site and the old. When you are no longer receiving email at the old account, you can cancel it.