The first time I experienced what I am posting, I restarted MusicBee with Auto DJ no longer set for playlist showing the playlist I last used. That was why I started the thread titled settings being lost. The playlist still existed; however, Auto DJ had library checked off again. This seemed odd, so I started this thread to question the behavior.
When I started to reproduce the issue taking screenshots, I restarted MusicBee with the same playlist selected under Auto DJ and Auto DJ was turned off. There is probably additional ways to recreate MusicBee resorting to library, but I did not spend a lot of time figuring out how that happened.
Additionally, I had noticed that a newer version of MusicBee was available. I did update to the latest available (moving from version 3.3.7310 to version 3.3.7367).
To answer the question of how I continued playback, I simply clicked on the Play button in the lower left of the Now Playing view. I have been doing this for as long as I have used the application. The play button here resumed playback exactly the way it was left prior to closing MusicBee. Auto DJ was still enabled, my playlist position was intact, and everything continued as before closing the app.