I would first make a backup of your current cart files.

Then I would copy all your edits to the classic folders to another set of folders that you create with a name for a template that will become yours and then edit the template_info.php file to reflect this name change.

Then in the admin > tools > template selection > you should see this new template > eidt > select it and update

admin > tools > layoutboxes controller > first reset and correct the sidebox display.

now you can try a FTP - complete reload of the cart files with out losing your edits.