I am trying to download this shapefile and get it into a format that ArcGIS Online will accept. The shapefile is primarily made of .adf files.
As soon as you see
*.adf files you are not looking at shapefiles.
Instead you are looking at either:
- vector data in Esri's old Coverage format
- raster data in Esri Grid Raster format
Neither are listed as supported formats at What can you add to ArcGIS Online? so to use them you will need to convert them into a supported format.