Another update, just in case this thread shows up in search results.

The back pain didn't go away with the longer, slower route, so I ended up getting an ergonomic evaluation (yes, there is such a thing) at my office. The lady who came took some measurements, concluded that the desk was the wrong height, monitors were at the wrong height and I needed to get a standing desk - one that allows me to alternate between sitting and standing. So I am typing as I stand right now, and my initial impression is favorable.

If anyone is in a similar situation, just build your own standing desk with a stack of books, and see if that helps. If you have a laptop, you will need another keyboard and mouse to make sure that your hands are positioned correctly when the screen is level with your eyes.