Carbon button in distributor cap broken off, melted

This is caused by having a gap between the rotor and the cap. Over time it starts arcing and eats away at the cap and rotor. Just replace the cap and rotor and be sure there is no gap between the two.

Exactly.
That type cap needs the rotor with the springy type contact that sticks up and contacts the cap button.
That is a skip white style HEI electronic distributor and as a matter of fact I have one of those here that the same thing happened to.

One thing about those is some were shipped with really shitty rotor phasing.
This one was so far off the engine would miss until the mechanical advance would start coming in and get the rotor closer to the post it needed to fire.
(Which I'm sure didn't help the cap burning problem any)

I had to redrill and move the pickup base plate about 1/8 inch to make it fire where it should.
Short version is it ran like **** at idle and low rpm's.