that is in settings>general. I encourage you to review all settings and account settings, as there may be several settings you prefer to change. welcome to thunderbird.

I chose the name only box, but the actual email address still appears in new messages.

if in your addressbook, there is also a box to check to just show names. The feature I mentioned in earlier response is recent, and there may be other considerations. you might try existing and restarting thunderbird to see if that affects it.