OK so the battery pull didn't help so you can try some other things that might work. It's hard to tell if there's a hardware or software issue so into won't hurt to at least try this as it
might work:
-- Clear the system cache partition. The system cache partition is separate from the general user partition, where all your files, data, and settings are stored, so this won't affect your data. You need to reboot your S3 into its Recovery Mode to do this. With your phone powered off, simultaneously hold down the Volume Up, Home, and Power buttons. When your phone vibrates you can release the Power button but continue holding down the Volume Up and Home buttons until your S3 is running in Recovery Mode. Use the keys indicated to navigate the text-only menus and select the 'wipe system cache' option. This will only take a few seconds and then just reboot your phone to see if this makes any difference.
-- If wiping the system cache didn't help, you could try to re-flash the factory ROM. Re-flashing the ROM is basically reloading the Android operating system. This link has directions on what to do, and keep in mind it's really important to download the ROM for your specific model of S3:
http://www.technobezz.com/flash-stock-rom-via-odin-samsung-galaxy-s3/
-- From what you've described I'd be leaning towards some kind of hardware problem. If it's something like the graphics chip on the motherboard, you need to weigh the costs of repairing your S3 or just buying a new phone. The S3 line is four years old now, as far as smartphones go that puts them in the 'geriatric' category. You could actually buy a refurbished one from swappa.com for less than the price of parts and labor to repair one:
https://swappa.com/buy/devices/samsung?search=galaxy+s3&type=phone
One thing you might want to try if you don't mind digging into your phone is to re-seat the display connector that's attached to motherboard. It might have become loosened over time. If you follow this guide on removing the plastic mid-frame that will give you access to the connector:
https://www.ifixit.com/Guide/Samsung+Galaxy+S+III+Plastic+Midframe+Replacement/12510
This jpg shows the connector (highlighted in red). Just pry it off and then press fit it back into place.
Anyway, let us know what happens.