The best to do it is to transfer all your content + databases + emails + crontabs + any other misc settings first to the new host.
Then you should ask them for a dedicated IP that you can test your moved sites with. Once everything checks out ONLY then update authorative name servers to the new host.
Since early September NS changes take 5 mins, and you can realistically expect traffic to switch over within 4-6 hours, so its important to verify the new destination is ready to accept traffic.
Check out
www.RealityCheckNetwork.com -- we do account migrations for free and have had very good success in the past.