hello,
I have a good bit of experience moving sites from host to host and can start on this today for you if you like. ideally, this would be done, as you mentioned, by backing up the old site, uploading it to the new server, and going from there.
What control panel do you have, if any, on these servers?
If you'll look at my profile (click the link directly, don't go by what they show in this specific bid), you'll see 100% project completion, 100% positive (5 star) feedback, and a link to my own site where I've done the same for better than 10 years now. Your project will be completed with the same diligence and persistance. Your task will be completed within a 2-3 day period of time, and be given everything needed to be done right.
I look forward to hearing from you and working with you on this project.
Tom Whiting