Hey @lancesalyers,
Our theme is unlikely to cause this issue with your category page as it does not have a function to change a post’s category so it won’t display in a category archive.
But to make sure if our theme is causing the issue, we first need you to clear all caches and temporarily deactivate your caching plugin. I see you’re using WP Rocket. Also temporarily deactivate all other third-party plugins then, switch to the default WordPress theme like Twenty Nineteen and check if your posts will display in the podcast-episode category. If the issue is the same in the default theme, there might be a setup issue with your posts and/or permalink. Please double-check the category and also try resaving your permalinks by going to WP Admin Menu > Settings > Permalinks and just click the Save Changes button.
If that can’t be done in your live site, please copy your site to a staging server and do the troubleshooting there.
If the issue does not persist in the Twenty Nineteen theme and with all third-party plugins deactivated, please give us your staging site’s WordPress Login URL, Username and Password and also FTP access so we could investigate.
Thanks.