Any help would be greatly appreciated!
The User Menu applet from the top panel was removed in 12.10, therefore by default, it does not display the logged in user’s name around the right-edge (in the system tray area) of the top panel.
Instead, the "cog wheel" menu at the top right corner, displays a list of users to quickly switch sessions.
Thus, you don't need to do anything to remove the logged in user's name from the top panel.
Now, if you've tweaked your installation and used Dconf Editor to show your user name on the top panel, just navigate to apps → indicator-session and uncheck "show-real-name-on-panel".
From the terminal, you can show the logged in user's name by running command below:
and this to remove the user name: