I broke the screen about 6 months ago and the cheapest option I found was a refurb from Gamestop.
...
Both phones are 16GB
You could have just gotten a flaky, crap refurb unit from Gamestop. They're so low on my list of places I think of going to for a used cell phone, they're literally not even on my radar. (Wait, Gamestop sells phones? Whut.) Seriously. According to your post, your problems literally didn't start until after you switched phones, so stop thinking it's the software. My fondness for Android may have decreased considerably over the past year and change, but your symptoms don't describe the OS flaking out, they describe hardware problems. The "refurb" was probably dropped a lot, and broke a few soldering connections and traces, and the motherboard is likely damaged. There might even have been some moisture damage. No telling.
The problem with your first phone was it had a broken screen. Your solution should have been to replace the screen, not the phone. The good news is, you can probably do that now for very little extra money because you now have a flaky S3 that you can poach a replacement screen from for free (
excuse a cheap set of tools) - though technically you'd just be swapping motherboards at this point.
You could probably even safely stop after step #22 instead of swapping out the camera fiddly bit since everything should already be there on both. It's not really like you have much to lose at this point.
That said, if you're still too intimidated to try a parts replacement or you find it doesn't actually fix the problem, then you should be shopping for a "new" phone. Is there any particular and specific app(s) for Android that you absolutely must have? I gotta be honest with you, if you can ditch Android, do it.
Opt instead for a Windows Lumia phone that can run Windows 10 Mobile. You didn't mention which carrier you have, so I can't recommend a specific model to purchase, but if you're with AT&T/T-Mobile, the Lumia 640 (AT&T RM-1072 or T-mobile RM-1073, carrier unlocked) is an excellent value currently. It is a phone that looks, feels and responds like a $300+ mid-range flagship phone
for well under $70 practically new that plays well with everyone else's cloud services (both Apple and Google).
Don't listen to the "app gap" nay-sayers (
check for yourself), and
don't listen to the "Windows Phone is dead" knuckleheads, either. It's a good platform, fast, stable, really easy to decrapify the phone and remove all the useless carrier software,
and hardly abandoned... and this is coming from a long-time Linux using, Microsoft hatin', scruffy IT neckbeard. I like the Windows Mobile 8.1/10 platform so much, I've abandoned my older physical keyboard phones (Blackberry and Nokia Symbian) for a Lumia. It's a phone that combines all the "right" parts of both iOS and Android into one device that actually has a consistent user interface, and can actually be set up to use
barely any mobile data at all without sacrificing usability. Battery life is also great for a smartphone.
What can I say? Satya Nadella has been good for the company.
EDIT: Forgot to mention that if you're really wanting to upgrade the Android build on your S3, go with
Cyanogenmod. Bit more mainstream and actively supported. But I would do this on a functioning S3, and not exactly expect it to "fix" what ails the current refurb you're using.
