So I called Sears today. I ended up on the phone with a pretty angry representative who didn't actually want to help me. She put me on hold for a minute, then all of a sudden the phone was ringing and I was on the line with someone else. I was unaware that I was being transfered, and this new rep didn't know anything about my situation. I asked what department she was in and she said that she handled "escalated service issues." I told her that this was exactly what I needed. Her name was Wanda. I told her the story of my washer and everything that had happened so far. She was pretty much appalled. She got on the phone with someone at the Sears store where we had bought the washer. She came back about 5 minutes later and told me that all I needed to do was to get to the store with my receipt, documentation from LG's servicer showing the attempts to repair and they would work on getting me a replacement.
Finally, someone wants to take ownership of this issue.