Based on Hoops last post, I just remembered that a friend of mine had once inadvertently put the later '37 inner/terminal caps in his '36 distributor and his engine ran or tried to run very poorly. It is possible that you have the reverse scenario, '36/earlier inner caps in a '37 distributor. It is now sounding a lot like something is not right with that rebuilt distributor. If you only had a known good '37 distributor that you could put on temporarily to try it out.
When this '37 finally gets fixed and runs well, we should all go and have a beer together.
Edit note: Well....based on Dick's last post my friend did the impossible....I guess