Is it possible to select either Fill or Stroke and move one or the other independent of each other? I can't figure out how to do this.

No, that's not possible. But I know what you're trying to do (I think) and it's quite a common technique in Inkscape. What you probably need to do is select the object, then Edit menu > Duplicate (or button on tool control bar). Next you will remove the stroke from the original object, leaving just the fill (unless you still need it to have a stroke later). And then remove the fill from the duplicated object. Now they are 2 separate objects which can be moved independantly

Edit
Just wanted to add -- be sure to check out Help menu > Inkscape manual (requires internet connection) and Help menu > Tutorials (internet not needed). Those tutorials are especially well-written for new users

Hey. Thanks for the reply. I tried this method but the Fill always extends into the middle of the Stroke and not along the outside edges of it. The only other option that I could think of is to duplicate the stroke, change the Stroke to Path and then use Difference to leave a cutout of the Fill. But changing the Stroke to Path brings up yet another issue mentioned here.

Ok, yeah I just replied to the other topic a little while ago. In that situation, where both parts are partially transparent, you would need the edges to meet precisely. I'm not sure if that's even possible. Let me know if my suggestion to make those pieces shades of opaque gray would work, rather than partially transparent black. If not, maybe someone else knows a way to make the edges meet precisely, because I don't.

I think the "problem" is that filled object is below the object that has stroke only.

Use Shift+PageUp and Shift+PageDown to move objects back and forth.