What are these marks on the Romex wire in the wall?
That is tar from the old insulation, that's why it is sticky. I don't know why or how it happens, but old insulation had tar in the paper, and it is very common.

It is not scorching or melting from overheating.

Take a utility knife and try to gently scrape some of the black away. You should see it's just on the surface. It's nothing to worry about.