I'd like to thank everyone for their suggestions. I determined that it was in fact a power button issue.
My "FIX" (for anyone with similar issues):
I removed the logic board from the phone and flicked the button up and down and that helped a little with the button's hypersensitivity. Because a slight tap would make the button "stick" and power off, I took apart the phone and physically removed the plastic power button. I could hit the button if necessary with a small screwdriver or my watch strap thingy, but I did my best to keep the battery from dying out. As for dealing with a lack of "lock" button, I installed an xposed module that let me remap my volume up button as a lock button.
My mother recently upgraded her phone and gave me her S3. Unfortunately it had a cracked screen. I removed her logic board and swapped it out with what was in phone and reactivated her phone as mine.
Thanks for everyone's help, and hopefully this can help someone else too!