Gregory
Following is a sample of the data directly in ArcGIS Pro with the hillshade function applied (Ie in the native projection of the raster, Virginia NAD83 State Plane North)
The checker pattern that is seen is a result of the original data being resampled during the reprojection. Any resampling (including all reprojection) will cause artifacts in the data. I can not confirm but it would appear that nearest resampling was used (vs bilinear). It is highly recommended that source data is not resampled. If the data was collected in a specified projection, sampling and pixel alignments then is it best to keep it as it is and only resample if required for display. If resampled then use bilinear. The issue here is that the data is resampled and then applying hillshade will extenuate the artifacts. Further resampling will only make it worse.