Authentication as an administrator when I click the lock in System Preferences?

Ok, See if you can create a new administrator account by restarting the Setup Assistant:

1. Boot into Single User Mode: Start/restart your Mac. As soon as you hear the startup tone, press and hold ⌘ + S until you see a black screen with white lettering. (If you end up back on the login screen after a flash of the black screen with white lettering, enter your password and it will return to the black screen.)
2. Check and repair the drive by typing /sbin/fsck -fy then ↩ enter - as directed by the on-screen text.
3. Mount the drive as read-write by typing /sbin/mount -uw / then ↩ enter.
4. Remove the Apple Setup Done file by typing rm /var/db/.AppleSetupDone then ↩ enter.
5. Reboot by typing reboot then ↩ enter.
6. Complete the setup process, creating a new admin account.

Be very careful to notice the spaces in those Terminal Commands.

Once you've done that the computer reboots and it's like the first time you used the machine, except all your data will stile there. Your old accounts are all safe. From there you just change all other account passwords in the account preferences!!

Great news! :)

• All iMac models introduced in 2011, 2012, 2013, and mid 2014: Use a Thunderbolt cable to connect your iMac to a Thunderbolt or Thunderbolt 2 port on the other Mac. If the other Mac uses Thunderbolt 3, you can use the Apple Thunderbolt 3 (USB-C) to Thunderbolt 2 Adapter with your cable.

Target display mode requires macOS High Sierra or earlier on the iMac used as the external display. It does not work with macOS Mojave or later.

In Sys Prefs>Users & Groups, does it say admin under your name on the left?