Well of course after struggling with this problem for a week, I figure out the problem just after posting!
Turns out it was a faulty logic/T Con board. Something to do with the chip on the board overheating. I reflowed it by baking the board in an oven at 375 degrees for 8 minutes and the problem has disappeared. From what I've read, this is the correct solution to fixing the problem temporarily, but a heatsink and fan needs to be installed to keep the chip from overheating in the future.
More information can be found here: