I raised a ticket with Apple - started an Apple chat. They got me to a Technical advisor (1 up from the normal guys). I was asked to restore the phone to do the following:
- Backup the phone (I created 2 backups on my Mac - 1 + 1 archive just to be safe)
- Restore the phone to Factory settings (I did this via iTunes)
- Set up the phone as A NEW PHONE
- Play a few DOWNLOADED albums only
- If everything works as expected, THEN restore from the Backup.
- Restore from the backup
- Wait till all apps get reinstalled
- Sync the phone to get the iTunes music albums back into the phone
- Test
In total, I spent 2:45 hours... 30 min with the Apple advisors on Chat, 1:15 hrs for the Backups (2 backups for me), and 1:00 hr for the Restore. For me, this seems to have fixed the problem. I think the difference between the earlier attempts and this one was setting up the phone as a NEW phone before proceeding. I will update again in a week/10 days.
Hope it works for you folks too.