If the front-end is working then there's likely no need to wipe anything out. That would be major overkill.

Renaming the admin folder is a simple process ... a couple edits to the correct configure.php file with a reliable editor and rename the folder.
Applying the patch is also a simple process ... putting the 5 extracted files into the right folders.

Blank-page problems are almost always a result of a PHP error caused by using a bad editor, adding extra blank lines to the beginning or end of files, bad FTP uploads, incorrect edits to files thus causing syntax errors in the PHP code, etc.

Sometimes using the debug utility mentioned in the middle of this article can help: https://www.zen-cart.com/tutorials/index.php?article=82


Your post started out with talking about v1.3.6 and then jumped to talking about 1.3.8. I'm not clear about what version *you* are actually using. And, if you decided to just dump v1.3.8 files on top of a v1.3.6 install, well, you've got some more work to do. https://www.zen-cart.com/tutorials/index.php?article=98