This is a more difficult problem, but I can hopefully help. Unfortunately Finder doesn't seem to have a good way to fix this so you will have to use Recovery Mode, Disk Utility, and Terminal for some of it. This should be considered an advanced procedure. Please do not attempt if it is above your comfort level.
1) Make sure your back-ups are in order (good advice regardless) in case things go horribly wrong.
2) Reboot the machine and hold command + R to enter recovery mode. See here for an overview of macOS recovery mode.
3) Inside recovery mode, start disk utility
4) Mount your main drive. Usually it's called Macintosh HD. You may need to supply your password if it's encrypted. When done it should say Mount Point: /Volumes/Macintosh HD (or similar).
5) Exit out of Disk Utility.
6) Back in the main menu of recovery mode, in the top menu bar, select Utilities->Terminal
7) In Terminal, enter: chown root /Volumes/Macintosh\ HD/etc/sudoers
8 ) In Terminal, enter ls -l /Volumes/Macintosh\ HD/etc/ | grep sudoers
It should look like this:
(paying attention to the first line - where it says -r--r--r-- 1 root wheel <a number> <a date> sudoers) Ignore the "sudoers.d" line.
Note that if your drive is called something besides "Macintosh HD" you will have to substitute your drive's name here when necessary.
9) If it doesn't say -r--r--r-- you may want to save yourself trouble and type: chmod 444 /Volumes/Macintosh\ HD/etc/sudoers
10) exit terminal and recovery mode and reboot as normal.
This should fix that immediate problem, but you'll have to let us know how it goes and if there are additional problems that show up after that.