Hei /6 won't shut off?

Great looking cars!

The slight stumble might be something else. I see ground connection at coil bracket, but uncertain in seeing ground at HEI module other than mount bolts. Aluminum forms an outer oxide layer that is transparent, and a great insulator. One of the mounting locations on the HEI module has a brass surface, it is the HEI ground. A wired connection to ground may help.

The two wires that go to the pickup coil in the distributor can pickup noise due to proximity of nearby HV coil wire. The noise can result in pemature spark, making the next spark low on energy. It is best to make that a twisted pair, short as possible, and away from coil to distributor cable and plug wires. Polarity is also important. The best check is to set crank at where base timing is set, then open distributor cap and view relation of reluctor tooth, pickup coil prong. They should be nearly alligned, but the reluctor is just passing slightly. If the alignment is between teeth, the pickup coil wires need to be reversed and timing reset. It is always best to static time engine to get one running. With all that, incorrect polarity more likely results in more degraded performance than slight stumble. The problem could be vacuum advance, or carb issue.

An e-core coil is less attractive, but puts out a better spark due to greater electro magnetic circuit efficiency.