I'd suggest to copy the whole osTicket directory first - so you have a copy pointing to the same database since the ost-config.php is the same.
Second, download ONLY the changed files (the 10 files) to the copied osTicket directory.
Third rename the existing files and copy the new/patched files to the locations of the old renamed files.
Now browse to the copy of your osTicket directory in the web browser and check if the issues are solved.
Last, when everything is fine rename the old osTicket installation directory so e.g. "osTicket-old" and the new directory with the fixed files to "osTicket" or whatever your osTicket directory is named on the server.
We do btw use a symbolic link to the real directory of osTicket on our server, so we easily switch between osTicket core files by updating the symbolic link.
Cheers,
Michael