Anula,
Ahh. I see.
Make sure you are not lifting your knees in half headstand. Keep them into the chest, hanging from the tailbone. Extend the elbows strongly into the floor. That keeps all your weight on your arms and the serratus muscles and lats engaged. If you are not able to support your weight there, then yes, more dolphin pushups. If you find the pushups aggravate your condition, then shorten the stroke to almost nothing and go slowly. Feel every degree of movement with the breath. Don't slump. Stand tall in the shoulder.
Don't be concerned about when you will be ready for headstand. It will come. Take the time necessary. You won't be expected to do anything you can't do at TTC, but you will learn how, so that when you are ready... It's what I meant about the difference between learning and practicing. You see? You can perform the headstand without holding it for purposes of taking instruction, but then just come back down. That way you will be seen and your technique confirmed, but you won't be hurting yourself. Besides, if you practice half headstand properly until February, there's a good chance you will be ready. By the time you leave TTC, for sure.
Good luck! You're going to do great. You're on the path.
Tell them all Siva says hello.
Om namah Sivaya,
Om Shanti,
[ ( + siva + ) ]