Does the Apple Podcast app no longer show badges for unplayed episodes?

I don't know what you mean by "reset". It generally refers to a forced restart:

While your device is connected, force it to restart. 

On an iPhone 8 or later: Press and quickly release the Volume Up button. Press and quickly release the Volume Down button. Then, press and hold the Side button until you see the recovery-mode screen.

On an iPhone 7 or iPhone 7 Plus: Press and hold the Side and Volume Down buttons at the same time. Keep holding them until you see the recovery-mode screen.

On an iPhone 6s and earlier, iPad, or iPod touch: Press and hold both the Home and the Top (or Side) buttons at the same time. Keep holding them until you see the recovery-mode screen

I'm suggesting a full restore to factory state (not restoring to a back up):

Restore your iPhone, iPad, or iPod to factory settings - Apple Support

Which one did you try?

Just updated to 12.1.2 and still not showing the ability to add the badges notification for Apple Podcasts. I wonder if it’s my particular phone, or carrier.

The carrier has nothing to do with whether or not your phone shows badges. If it were me, I'd back up the phone, restore it as new and test.

My apologies for the confusion. I did a full restore of the iPad Air 2, and set it back up as a new iPad. I also updated to iOS 12.1.1. This did not resolve the "no badge notification" issue for my Apple Podcasts app. See image below.

Regarding this particular issue, I've also contacted Apple Support. The end result was that there was no resolution. The only thing the CSR could tell me was that "this feature may not be available anymore for this app".

I'm sorry. I have no other ideas. All I can tell you is that, I have iOS 12.1.1 and the option is there so, the CSR is wrong.

Do other apps have "Badges" as an option? The only other thing I can think of to suggest is that you try another podcast app. I really like Pocket Casts. I haven't use the Apple app in years.

You can get notifications for new episodes. Settings>Notifications>Podcasts>Podcast Notification Settings. Make sure Badges is turned on in Settings>Notifications>Podcasts.

I'm running 12.1.1 and the switch for badges is between Banner Style and Options. Try updating?

I have Overcast, which I got to sync podcasts to my Apple Watch (before Apple offered that option). But then I discovered I almost never listen to podcasts on my watch.

Good luck!