OK, I just figured this out, but to answer your question first, Jeff, Mandrill appeared to be getting no notification whatsoever that the app wanted to send a message unless the sent email it came from the email test utility.
Oddly, setting the DB hostname (from the IP to the proper name) in the application conf and restarting sidekiq is what was required. All appears to be working properly now. 
You guys might consider having your email test form use the same facility to send mail as other parts of the app, since it’s a bit confusing when the test works but actual emails from the app fail. Just sayin’. 