Porting/Moving site to another and upgrading - questions about data and template

Hi all, I've not long moved into a new job where they've already had a zen cart site set up, I've now got the hang of things but they now want me to move everything from the website they're using at the moment, to a newer site..

Okay, so at the moment they're using https://www.wetndrywall.com/catalog, and they want me to move all that stuff to https://www.wetndrywall.com/shop but keeping the template on the .com/shop site.
both sites are on the same ftp, so I'm guessing it's just a case of backup then restore, but I think that will lose the template!?

Thanks for looking and sorry if it's in the wrong forum.

Edit: I've had another look and the two sites are differant versions I think, the site they're using atm is Zen Cart 1.2.5, Database Patch Level: 1.2.5 and the one they want to move to is Zen Cart 1.3.7, Database Patch Level: 1.3.7. How confusing!