When I moved to a new host, I placed a comment in the HTML at the new server so I would know for certain which place I was getting the content from.
Also, you should be able to find some free web proxy servers around the world you can use to test which server is delivering the content. This will help with DNS caching issues as they will not have your old IP address cached whereas your computer or ISP may.
Free Web Proxy 1 - UK and possibly USA IP addresses.
Free Web Proxy 2 - Germany and Seychelles IP addresses when I tested.
And you can always examine your visitor logs on your old server to see when traffic has died off. That will give you an idea of when you can shut your old server down.