iPhone 4s Randomly Rebooting Post 8.3 Update

The onset of the problem:

Updated to iOS 8.3 over the air through wifi download on a full battery of 96%. Started it before I went to work. Forgot about it. Got to work 40 minutes later, iPhone at Apple logo. Didn't think much about it. Put it next to the computer, did something else. Noticed it in a constant reboot cycle because it eventually drained the battery and died.

Attempted to halt the reboot process by pressing home and power buttons simultaneously after it had enough power to boot again. This did nothing to correct the reboot cycle.

Attempted to get in to DFU mode. Was successful. iTunes was able to recognize it and aware that an update was required, or possible full reset. Attempted to complete the update through iTunes. The updated appeared to be successful, yet entered in to the same reboot loop as before.

Again I attempted to halt the reboot process by pressing home and power buttons simultaneously. This did seem successful and actually loaded the lock screen after which it rebooted again. It loaded the lock screen again and I was able to unlock it. I got this error: your selected cellular network is no longer available. You can choose a different network in settings.

I connected to wifi and the device rebooted again. Got back to lock screen, home screen, and eventually after what seemed a lot of toggling between edge, GSM, 3G, and nothing, it eventually locked on to my carrier. Back in business (I thought).

Tried to get iTunes to connect, couldn't connect. When it did start to connect, everything was grayed out. Eventually after several more reboots it finally connected, and I was able to get a backup off of it. Took a few interrupted sessions, but I did get one.

Now it appears to reboot randomly. When I plug it in to the charger. When I plug it in to the computer. When its been plugged in to the computer, already rebooted, and reboots again for good measure. Same when on the charger. After it has been on a while it will just reboot when I haven't touched it. Sometimes it goes through a couple cycles, but eventually comes back. Just recently it locked up when I was on speakerphone and tried to use the keypad. After pressing the show keypad button, it was half highlighted/half not, the UI became unresponsive to touch, and eventually rebooted.

My only other thought is to do a factory reset on the phone and start again, but I really don't want to lose a lot of my app data. Any one else have or had something similar going on?