Well after long weekend of fridge drama, I am happy to say that I have nice squeaky clean fridge that now works properly. Thanks to this man...
Evidently the water to the ice machine was running continuously and once it overflowed the ice container, it froze the bottom of the shelf to the bottom of the freezer.
By letting the whole thing defrost, we were able to fix the problem. (I use the term "we" here loosely - see above picture) Apparently there were a few more steps involved, but this is what I remember!
I spent most of the time washing and sanitizing every square inch of the fridge. I knew it needed a bath, but I had no idea how gross a fridge could get.