I got it. Let’s try to fix the issue with the BB first.
Please deactivate everything related to Pods, since we don’t need it anymore.
On the front end, if you keep your old modules / shortcodes, then the custom fields with Pods might be still outputted correctly. It’s not a bug, it’s just BB can still retrieve data from custom fields even when the plugin is deactivated. (Of course, the data is still in the database, so BB is able to retrieve it).
Regarding fields that you created with Meta Box, can you check if you installed and activated the Meta Box – Beaver Themer Integrator extension? This extension helps you to output the custom field easier. See the screenshot on the plugin page and let me know if there’s anything unclear.