You say that you have two clients installed - the 11g client and the 12c client. Are you certain that you are using the 12c client? Because if you deleted your tnsnames.ora file under the 12c client directory, your error message should have changed. This leads me to believe that you are likely not using the correct client. In a comment you said:
C:\app\oracle\product\12.1.0\dbhome_1\NETWORK\ADMIN. The first
contains the 11gR2 client install while the latter is the actual
Oracle 12c database. When I make a change to the tnsnames.ora under
the client directory, it gets reflected in the Windows ODBC screens.
This makes me think that my suspicion about multiple client issue is correct. To test this, open a new command prompt window and issue the following command.
Where [DATABASE] is the TNS entry you would like to test. This will tell you which client your system is defaulting to. If you are in fact using the 12c client, you should see something similar to this immediately following the tnsping command:
TNS Ping Utility for 64-bit Windows: Version 220.127.116.11.0
If you do not see Version 12, you are using the 11g client. To fix this, you can modify your PATH system variable but you will need administrative access. Search "view advanced system settings" in the windows search bar, go to the "Advanced" tab, then navigate to Environment Variables. Scroll down the System Variables list, find PATH, and then modify the string so that the reference to the 12c client appears BEFORE the reference to the 11g client.
To make sure it's configured correctly, try another