Just wanted to post an update on my project.
As the good folks here advised, I ripped out the old wire you saw in the pics. They were not as hard to trace as I thought. They actually went down some old molding that I didn't even realize was wire molding. What I found inside was this.
Quite frayed. Caked on top of the frayed portion was this. It was a bit crumbly, like Play-do that had dried.
I'm not sure what whoever installed this was thinking. I'm just grateful it never turned into an electrical fire. Anyway, all of that went down the garbage chute. I went out and bought Romex 14-2 and snaked that thru some new wire molding to my new spot. A new electrical outlet with a proper ground went in.
And I now have my outlet where I wanted it and it works great.
Thanks to the people in this forum who stopped me from doing something stupid!