Why are my distributor cap terminals burnt?

I'd take a look at the vacuum pods and check function. Long story short, the rear pod has a "gummy" or "sticky" feel to it. It does move under vacuum, but not smoothly and does not return to position easily. The front pod is very different. It's smooth and I can depress the "arm" that goes into the pod easily, and it returns to position on its own.

I disconnected/plugged the vacuum line to the rear, and HOLY SMOKES, it's like I'm driving a different car. Acceleration using a heavy foot is night and day different. This was the first time I've been able to put the pedal on the floor and hear the transmission downshift -- before it wouldn't even get to the point of downshifting before sputtering and misfiring. However, it still stumbles at higher RPM, so I'm going to pull the distributor and take a look at the aforementioned items.

I have so many darn threads going on about this car that I wish I could consolidate into one.
Assume you're talking about the outboards carbs on the 6-pak.
That's good. Now you've isolated a problem. Leave them disconnected for now. It will work fine on two barrels, certainly for street use.