I spent a couple days trying to find someone with a backhoe that could come dig up the pipe so that I could fix it. I didn't want to have to dig it up by hand. These are the Rocky Mountains. There is no soil. But, no one was available to dig a hole for me for at least a week so I decided to hire a friend and dig the hole by hand.
The plumber had marked the location of the offset pipes, so using a pickaxe and a couple of shovels, we started digging. It was hard work. One of us loosened the rock and soil with the pickaxe, then the other cleared away the loose earth. We had to dig down three feet to get to the pipe and ended up digging a hole four feet wide and six feet long.
When we got down to the pipe, the source of the problem was apparent. Two pipes were connected using a flexible rubber coupling and the pipe closest to the house had sagged several inches. The pipes never actually disconnected (the cut in the coupling in the pictures is from me).
A flexible coupling is really the wrong thing to use for this application, but the best I could do was reset the sagging pipe and replace the coupling. I supported the connection with a four foot long 2x6 that extended on either side of the joint and packed the dirt underneath it with a sledge hammer to keep it from settling in the future. The toilettes are flushing like normal again and hopefully we don't have to deal with this issue again.
The job wasn't over after I fixed the pipe. I've always believed that "the job's not done until the cleanup is done," and there was still a lot of cleaning to do. Since we'd had to plunge backed up toilettes filled with sewage water a few times and bacteria grows readily on surfaces contaminated by sewage water, I had to clean both bathrooms and the kitchen thoroughly. I spent two hours bleaching every single surface, including the shower, sinks, toilettes, and floors. My eyes and throat burned by the time I was finished.
Now, the toilettes work and the cabin is cleaner than it's ever been.