Following the steps in this question has yielded some pretty good results. I have just one issue, which was also experienced in the original question. The edges of my height map are oddly high.
I loaded the DEM in GIS software, and here are no odd values; QGIS tells me that those pixels are marked as "no data". This leads me to conclude that the issue is happening in Blender. The answer in the original question tells the questioner to clean up his height map, but how is this done, especially when the edge-errors do not appear to exist in the DEM.
Edges in Blender (camera rotated to make edges more apparent):
No edges in QGIS:
