Background Eraser Tool Leaving "Wisps of Grey"

Hi Zachary.  I'm not exactly sure what you mean by the whisps of grey and white whisps all around the face.  Can you be more specific.

The Subject selection tool is not perfect, particularly with the variations of similar tone around an image such as your original and difficult areas like the strands of hair.  So, when I take your image and apply the Subject selection tool, there are clearly areas that are under and over selected. 

But it seems to me that you must have used another selection tool to remove, for example, parts of the armor that have been selected.  If you click on the Refine Edge tool, there are variou viewing options that show your selection more clearly.  Here I have used the red Overlay option:

I further assume that you inverted the selection so that you would only remove the background.  So, can you please explain exactly what you did to clean up the selection before using the eraser tool. 

And rather than use the Eraser tool to remove the background, if you have the subject selected, you can press Ctrl+J

 to copy the subject to its own layer.  You can then add your new background under that layer.

Note that you should always duplicate the original image (also by using Ctrl+J without any selection) before editing any image.  You can then hide the original (by clicking on the eyeball icon).  That way you can always go back to the original if something goes wrong with the edit.

So, bottom line, I don't think you are doing anything wrong.  You just need to pay more attention to what is actually selected.  The Refine Edge options will help you see that more clearly.  Here is a link to the Help articles about selections that you may find helpful.