I agree with Dave's last comment. I think the best way to do this is to install a local server environment. I used
Joomla Pack
to install a replica of my site on MAMP (for Macs). Then I could install a new template get it looking exactly like I wanted, then all I have to do is record what modules need to be moved, zip up the template folder on my local machine, and install it to my live site. I usually do this at night when I know traffic will be light.