I finally figured out the issue. Seems the template I was using, Pinboard, was calling a couple scripts that were breaking the podcast. The scripts were wp-mediaelement.js and mediaelement-and-player.min.js, located in /wp-includes/js/mediaelement/. I commented out the following in my functions.php file:
$('audio,video').mediaelementplayer({
videoWidth: '100%',
videoHeight: '100%',
audioWidth: '100%',
alwaysShowControls: true,
features: ['playpause','progress','tracks','volume'],
videoVolume: 'horizontal'
});
There were two calls to it.
The player seems to be working properly now.