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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): Edges

No edges in QGIS: no edges

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    $\begingroup$ Are you using linear interpolation for the texture? Have you tried with Closest interpolation? Can you provide the source image file? $\endgroup$ Commented Jan 16, 2021 at 12:04
  • $\begingroup$ Thank you! This and changing the image extrapolation fixed it! $\endgroup$ Commented Jan 17, 2021 at 4:13

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If you mean distortion around object's edge, change image extrapolation to Extend.

Displacement Modifier > Texture Properties > Mapping

enter image description here

... in case you use Subdivision Modifier > Adaptive Subdivision:

enter image description here

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