Having the same problem. Not sure what the hosting service is because they won't give me cpanel access. Anyway, I have Akeeba Admin Tools installed and I click on Fix Permissions and it fixes it but only temporarily. I can make changes, save them, but after some unknown amount of time I look and it's back to unwritable.
:arrowu: Have you tried, or are you able to try the fixes posted by Kat in this thread?
Mainly, making sure the FTP layer is configured properly (or disabling it), and the other edits in the /administrator/components/com_templates/controller.php
The FTP layer setting should be available in your Joomla Global Configuration, the other one, without CP access, I'm not so sure, but if you have FTP access you may be able to edit the file.
This is usually related to the server. You should have them set as 755 and make sure apache has ownership of it. You can set this via FTP or get your system admin to do it.