Thanks Joe. Just to update for those having a similar issue, I actually had to have my phone replaced today at the Apple store.
Despite my reticence, I was having so many problems with TID I decided to delete my right thumb prints that had been working since October and tried to start from scratch. When a new right thumb print didn't work right out of the gate I knew I had a bigger problem. I decided to take the nuclear option and restore the phone (from DFU mode no less) and set up as-new. So, completely clean install of firmware and iOS 7.0.4. Still had intermittent TID, with "try again" until passcode unlock 3 out of 4 times. FIgured it must be a hardware issue now.
Went to the Apple store and was given a replacement without much question after I told the Apple Genius the phone was already restored and set up as-new and was still failing, which I was able to demonstrate for her. She said she had seen it a few times before, but didn't give a lot of detail. I forgot to check the manufacture date on the box, but it shipped with iOS 7.0.3, so it's definitely more recent than the one I had. Serial number starts with DN (but so did the last one I had, so not sure that makes any difference).
Set up TID right there in store with single print (right thumb). Worked perfectly. Got home and restored from backup. Still works perfectly. I'll see how it works over the next few days, but perhaps I did have defective hardware causing the complete failures. Keeping my fingers crossed!