I find that the easiest thing to do, is to create a new empty folder in the testing server root directory, and then via 'Manage Sites' in Dreamweaver, go to the Testing Server settings and select that as the new 'Testing server folder:' (and reflect that in the 'Url prefix:' as well).
Then you test out your site by previewing all of the current pages (not the old stuff), so that what you effectively end up with is a clean folder with only the current stuff.
Keep the other stuff in case for a while and for reference, but at least it's all separated out and can be junked when the site has been thoroughly road tested live.
Also I'm not sure what you mean by still having active links. That would all depend on whether the old files were linked to the new, or the new files linked to the old.
As well as sending stuff 'clean' to a new temporary Testing Server folder, you could check all links site-wide in DW (Ctrl+F8) once you've cleansed the site files to spot any potential problems. The key, I suppose, is to keep the old stuff backed up until the new stuff is thoroughly proved.