I will guarantee 100% that your power button has failed on the board, and I myself have googled videos on Youtube etc... My s3 was in an Otterbox since day 1, never dropped and I had this same issue.
Nothing will fix it but replacing the power button. If you contact Samsung theyll charge you $50 if you ship it to them and they'll fix it, mines not new either.....
I however being the stubborn person, knew I could "fix" this myself and I did.
I took my phone apart, took the board off, and using a pair of tweezers, I literally kept slowly wedging the tweezer behind the button on the board, right behind the button, until eventually the middle part with the button came off.
I put the phone back together, and while holding Vol Down + Home, plugged in my wall charger and you'll get that screen to go into download mode OR you press Vol Down to restart, it'll restart your phone normally and Voila! It's fixed.
I have never had a problem since, however if your phone dies the only way to power it on is as I just mentioned. I use Quickboot from the play store on my phone and that's how I boot into recovery to flash my roms, or reboot my phone etc... bottom line, if you want you can do as I did, Or contact Samsung directly. You can also go into sprint and they'll let you know how much they'll charge to fix it as well, I was told $75 and I don't have insurance either.
Have a great day
1 other thing... If you have a multimeter, you can test your power button on the board, to see if it's "open" and you'll see it is in fact the power button/switch on the board.
---------- Post added at 11:43 AM ---------- Previous post was at 11:38 AM ----------
1 other thing... I bought a new battery too, a stock oem one directly from samsung. I couldn't find it and contacted them, they replied with a link to one and that did nothing for me.
Your situation is almost exact as mine, it's that whole "Sudden Death Syndrome" which is in reality, a failure of the power switch/button on the motherboard.
We never really grow up, we only learn how to act in public.