Help me to edit sudo file

I am trying to edit sudoers file by running the command #visudo. But it is not opening and error showing like 'it is read only filesystem'.

Than I changed the permissions of /etc/sudoers file to 640 and modified it(after I change the permission to 440). Than it is modified successfully. But sudo access is not working.

After investigating I found that in /usr/local/etc/sudoers file, the modified fields are not updated. Even it is not allowing to change the permissions of this file. I tried it as a root user.

Please help me to resolve this issue.

Did you modify it using visudo after chmod?

Man Page for sudoers (all Section 5) - The UNIX and Linux Forums

CAVEATS
The sudoers file should always be edited by the visudo command which
locks the file and does grammatical checking. It is imperative that
sudoers be free of syntax errors since sudo will not run with a
syntactically incorrect sudoers file.

Supposed your sudo|visudo commands use /usr/local/etc/sudoers,
then check its file system with

df /usr/local/etc/sudoers

If this is NFS then one must edit|visudo it on a machine with trusted root access.
Trusted root: the NFS server exported the file system with root=machine .
Maybe there is another sudo installation that takes another sudoers that is on the local disk?