What I want to do is simply transplant the existing application onto the new database. I'd be happy to simply copy the files over and hack the .conf files appropriately, or if it's better, can I use PG_DUMP to somehoe backup the old data and restore it to the new installation?
You are going to need to dump and restore the database(s). The data files are likely incompatible if you are jumping between releases, unless it is a maintenance release ( e.g. 8.2.0 to 8.2.1, but you should refer to the release notes to be sure).