Phone won't turn on, just vibrates when plugged in

Question

Last night, I plugged my phone in before bed.  It showed on the battery icon that it was charging, and the red charge indicator light was on once the display turned off.  However, by the time I woke up, the phone was dead, though still plugged in.  I couldn't get it to do anything, so I tried an old battery that I had.  Put the battery in, the phone turned on and showed a 47% charge.  Knowing that battery was garbage (which is why I got a new one; I just held onto the old one for emergencies), I put the phone on the charger and the battery icon showed charging.  Instantaneously, though (I'm not exaggerating, it was literally instant), the phone dropped to a 1% charge.  I quickly tried to turn the phone off to see if I could even get even a little bit of an efficient charge out of it, and the phone started power cycling.  I tried to stop it, but it turned itself off and on probably 20 times before it finally totally died (kind of reminded me of how sometimes an sick animal will go bat guano bonkers before it kicks it). 
Now, when I try to plug the phone in it just pulse-vibrates every 2 or so seconds instead of showing that it's charging.  It will continue to vibrate like that for a minute or two after I take it off the charger before it finally stops.  Won't turn on in any mode, plugged in or not - I do get a flicker of the opening "Samsung Galaxy" screen before it shuts off and goes back to vibrating.  I've tried to let it charge for 30-45 minutes, but no change.  Taken the battery out, smacked it a few times, tried to charge with my computer, tried different cables/chargers, nothing helps.  I have no idea if either battery has any charge at all at this point (though I suspect not).  I haven't damaged my phone at all - no recent/significant drops, no water - but of course I'm just barely barely out of warranty.  Am I toast or is there some way to fix this?

Answer 1

Thanks for your reply.
Unfortunately, since (as I clearly noted in my original question) my phone will not turn on, I can't confirm the phone software, though I've downloaded any and all updates that I've been notified of.  I have no idea if any apps updated immediately prior to the issue happening; it's likely that some update to something downloaded not long before the problem started since I plugged in the phone shortly after I got home for the evening and reconnected to my home WiFi, but I wouldn't have any way to tell you what it might be.  My phone will not boot in safe or recovery mode either, if that helps answer if an app is causing the issue.
I tried to take it to a Verizon store; they told me that they had literally NEVER heard of my issue before and so wouldn't be able to fix it and didn't even attempt to help, just tried to sell me a new phone, so that wasn't super helpful.  Since I had zero problems up to this point with my phone, and because I'm now not eligible for the early upgrade to the Edge program without the early termination fee because I don't have a working phone, it sure would be great to fix the one I have.  If anyone else has any advice, I'd appreciate it!