I've narrowed down the problem. At first I thought it was random or a fluke. Now I notice that every time I use the in-game map and start moving the cursor around or zooming in/out, my audio cuts out or sometimes stutters and then bugs out leaving only a constant buzzing sound.
While I don't know if it matters, this started happening while I was in Riften.
My headset is a Turtle Beach headset designed for use with the Xbox One. I'm using a Microsoft Xbox One controller with a wired connection. This has never happened prior to the Dark Brotherhood update.