I'm experiencing the same problem. Before upgrading to IOS6 I was listening to a Filmspotting podcast. After upgrade, I started playing it from the Podcast app. But I can still find the Music app also showing it Now Playing.
I switched to another Filmspotting episode and the problem persisted. My car connection is via a 30 pin dongle that broadcasts to my car radio.
The problem persisted when disconnected from the dongle and using the internal iPhone 4s speaker. I then tried a podcast that had several 10 minute rather than 60+ minute episodes. Those worked. Powering the phone off and on did not resolve the issue.
Now, I'm home, still on 3G. Now only the FS episode that has a duplicate in Music app is stopping. The other FS episode is playing just fine. The other podcast is playing just fine.
I think it's related to iCloud syncing because I notice that when I stop the FS episode from Podcasts app and run it in Music, the position is synced to where I left off and same when going back to Podcasts app. It also is probably related to large files.
It's quite annoying. I also don't like that the new maps app turns off podcast playing. I thought maybe running the maps app was the reason, but I think that would have cleared up when I rebooted.
Maybe if I figure out how to delete the episode from the music app it will be fixed.