[Q] Galaxy S III crashes, won't boot unless connected to power source

I took Verizon's NC1 OTA about two weeks ago. This weekend, Maps kept locking up, and now that has devolved into the whole phone suddenly dying. No power off animation, just the screen goes black and the phone is off. When I attempt to boot it back up, it'll do the same sudden death bit, sometimes without even making it through the Verizon boot animation. The battery had a strong charge the first time this happened - about 70%. It will fully boot if I connect it to a power source, but it will die again after some active use off the charger. As a result of this, I'm also unable to boot into recovery. I would ultimately love to flash some other ROM as I think I may have corrupted something when I took the OTA. Battery is new-ish (or, more likely, refurbished-ish) - replaced the phone via insurance back in December.

Some additional background: I had my phone rooted two OTA's ago. When I installed the second most recent OTA (I forget which, maybe MC1?), the phone obviously unrooted, but then I also got the "Samsung Custom" boot screen with the unlocked image, which I thought odd considering it was an official OTA.

In summary: I have a phone that is rather useless unless I'm perpetually near a power source. Has anyone had a similar experience, or does any one know what the heck I can do to fix this?

It's alive

So disregard the bulk of that post. Turns out the power problem was a bad battery. I popped in the battery from the old, smashed phone and it worked like a champ.

However, I'm still stuck with the problem of not being able to boot into recovery. I'll get the same Samsung logo with the unlocked logo and "custom" at the bottom, but then the screen will go black and nothing will happen until I let go of the key sequence, at which point it'll finally boot up normally. Thoughts?