I am experiencing something very similar, since updating to 16.4.

I have moved a way from automatically include method and am having to manually select podcasts to sync. I can listen to podcast this way. I can uncheck podcast I have listened to, check new ones and am able to re-sync. Using this method, the podcast stay on the phone only for a while. Then all of sudden when I open the podcast app, the podcasts start disappearing before my eyes, or I go into one of my station - that says I have podcasts in it before I go in - and in it, it shows no podcasts. They when go back to iTunes to re-sync to put them back on the phone and find all the podcast I checked to sync are unchecked. I have to re-check podcast to sync.

If I automatically include podcast to sync, I cannot get them to stay on the phone for very long (less than an hour) and have seen them disappear before my eyes in the app.