The "upgrade available" link just downloads the files.
You should probably go to osticket.com/download and grab 1.12.
You then have to upload the contents of the /upload folder in the archive to your server.
You can either overwrite the ones there or delete them all.
note: save the /include/ost-config.php because that's the only file that you need.
note2: if you are using the Attachments::Filesystem plugin do not delete your attachments folder.
note3: if you are running any plugins make sure you get new versions from osticket.com/download and put them in the /include/plugin folder.
if you deleted the site and put the new files there already (which you indicate that you did) then I suspect that you deleted and put the files in the wrong place.