Hi Chris,
It will let out a light cough and small backfire when it starts acting up at idle, but the problem mainly occurs under acceleration. The more I baby it, the longer it goes without acting up. Once it acts up under acceleration, it doesn't clear up until I let it set. It doesn't have to set long to clear up - maybe 15 minutes later and it will run again okay for a while.
After reviewing the play in your distributor shaft, and although play in that area could affect ignition operation, I personally don't feel this is the cause of your problem.
Is your Healey + or - grounded?
Have you checked the carburetor bowel levels as previously suggested?
I keep coming back to 3 disjoint thoughts:
1. Heat induced vapor bubble or debris after the fuel pump.
2. A ground fault resulting from an unintended consequence.
3. Aggressive ignition timing.
Bubble or Debris
The first is an air bubble or debris in your fuel supply, past your fuel pump, that is blocking sufficient fuel from being received in 1 or both of your carburetors. The multiple causes of an air bubble has been mentioned previously and could be causing a heat-induced vapor lock. Additionally, With HD8s, there is a chamber past the pin and under the needle that could be harboring debris related to dried fuel during a long storage. Contamination in this area is not common and, I believe, would only be caused by fuel left in this area to dry over an extended storage period.
Unintended ground Consequence
Have you replaced the OD dash switch. Why was the OD re-wired to allow it to function over all gears? As mentioned in a past post, the OD normally shares a main ground with a number of other key electrical elements it is not that uncommon for a bad ground to find a path to another grounding point through attached unintended circuits.
Advanced Timing
What is your ignition timing set to and how did you set it? Aggressive ignition timing could cause issues when also aggressively accelerating. This situation, combined with reasonable cruising richness, could easily become too lean under aggressive acceleration. Add a weak fuel flow to this scenario and you could produce the conditions described. Yes, this is a stretch but these factors could easily fall in place. It is reasonably easy to dynamically check timing and view the function of your advance. Additionally, checking and setting carburetor richness to a higher testing level is also relatively easy.
I bring these paths as uncommon possibilities and do not want to exacerbate your present level of frustration. However, once identified, I expect you will find your issue's cause will be relatively easy to rectify.
All the best,
Ray(64BJ8P1)