I just wanted to give a quick thanks to Mrsnork, as she has been able to help me the most with my TouchID problems. I was ready to take my phone back for a replacement when I started reading this thread. Thanks to Mrsnork, I found there were a couple of problematic apps and some low memory issues by looking at the diagnostics page. I uninstalled those apps, reset all settings, and restarted the phone. I set up a new touch ID, only once, on my thumb. So far it's been working great and it's been about four days now. Before, whenever I would set up TouchID, it would work perfectly for the first couple hours, and then greatly diminish after that to the point where a couple days later it was not worth even trying. Now I would say I get around 9 times out of 10 with dry hands, which I am happy with as long as it does not start to degrade again. If it does start to act up again I will first restart the phone and see if that solves it. But so far so good. Hopefully Apple comes out with a fix for this soon or they will be getting a lot of phones coming back to them.
For all those who haven't read her experience, start at around page 3. I don't think you need to do a clean install of the iOS on your phone. Maybe not even reset all settings, I'm not sure. I'd start with looking at your diagnostics for any apps that may be crashing, or low memory issues. Uninstall those, and restart your phone. Try your fingerprint again and see if it works. If not, erase it and start a new one. Wait and see if that works for a while. If not, then you may want to take the additional step of resetting all settings.