You can do it, but you'll have to overlap the two hosting terms a little.
You can upload the content to the new server, and set up the sites on IPs...make sure everything is working. Then, have the new host set-up your domains in the DNS to point to their corresponding IP. Then change the nameservers on your domains at your registrar (NetSol.com, Register.com, DirectNIC.com, etc) to match that of your new host.
It doesn't always work that smoothly, but that's what should happen.
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