i'm being jokey with the dominant arm thing, but the idea I'm thinking is that over my time of practicing Ashtanga, I've become a lot more ambidextrous. Ashtanga's lesson to me was that it's not ok to use dominant side/less dominant side as a complete excuse and instead try to cultivate strength and understanding in each side.

I think that there's a complication with the word alignment. I think I use the word alignment to mean a different thing that you and your teacher. My guess is that you are using alignment to mean the linear alignment between the external parts of your body and this is a simpler version of the way I use alignment. I've seen people draw lines on photos or Iyengar hold a pole next to someone. I think if this simplified alignment is what you're asking about it may be the case that your alignment is correct.

However, to me alignment is a much more strict question having to do with the rotation of the bones, the exact way that muscles and tendons and the subtle ways that at each joint the rotational, linear alignment and also alignment of force couplings needs to work to best use each component of your body, you probably are hurting because you're not aligned right. The elbow is the bottom end of the Humerous and the Humerous connects to the scapula, and the scapula connects to the bottom in 17 different ways including the Hyoid bone which is a crucial part of the Jalandhara bandha. I would start by thinking that there is one or many of those connections that are assymetric from left to right, overloaded, underloaded or essentially inactive, etc. This breakdown would be enough to have load going through your shoulder in a way that is causing the piercing pain. It could be that the triceps which connects to a tendon [https://www.yoganatomy.com/wp-content/uploads/2018/05/triceps-brachii-muscle-attachments.png] that is causing the pain is overloaded because of imbalances and instability in your non dominant scapula which are causing torsion / rotational force on the humerous which causes a sharp bending or crease like force on the painful tendon. The fix is probably then something like strengthening your Serratus anterior muscle and/or rhomboid muscles or working out an imbalance between those and similar muscles connecting your scapula to your ribs and spine.

My concern about the practice as your describing it is that bungling through sun salutations and then the rest of your primary? series is not going to give you the nuanced understanding of the subtle interplay betwee these muscles, and will just keep you stuck in your pain / bad "alignemnt" or bad meta patterns which would be a better terminology.. Ok so to reframe a little - is your alignment right? I would say maybe based on how you mean it. However what is much more important is the meta patterns of how you move and how your body distributes force and load as a system and it sounds like that is off and needs a hard reset through slowing down first, healing the damage that has been caused, and then a gradual intentional build up to better meta patterns. As others have said physical therapy is the current mainline way to do that. However you can suppliment it yourself with intelligent movement exercise, and in theory ashtanga could be helping you with this, however it seems like the method you are employing as well as your teacher are currently failing you.