Well, I was able to get my Podcasting app to work again, but it's not really a "fix" as much as a "hack". I downgraded my phone to iOS 8.4.1. and restored from a backup to get it to work again. I did have a hitch, however. I backed up my phone just prior to upgrading, like you are supposed to, and then backed it up after the upgrade, which was a mistake. Seems unless you've done a restore, backups are incremental so when you upgrade the next backup with overwrite your last 8.4.1 backup and make it impossible to restore onto 8.4.1 from it. I was able to use a time machine backup to copy in the last good 8.4.1 backup, but unfortunately that was from a month ago. I'll have to do some more work to recover photos and such from other backups.
I have to say I'm very displeased with the lack of help or support from Apple on this issue. Like others, when I contacted support about this, they claimed they have never heard of this issue, which I don't understand how they can still be claiming this after seeing all the folks from this thread reporting it, and then they tell you to do a full restore which, again from looking at this thread, does absolutely no help and just wastes a couple of hours of your time for no good reason.