I think your daughters behavior is very normal for whats been going on. A move, a new baby, and an illness can be triggers for this type of behavior to happen, and having all three, well her response is understandable. My daughter has always been very independent, never ever clingy to me, until I became pregnant with number 2. I included her so very much, she went to all my ultrasounds and OB appointments, she went to sibling class at the hospital, we read books and watched videos about it. She was still very needy and clingy and really, it did not stop until about a month after the baby came, I wish I had more helpful advice for you!
Also, perhaps it is just me, but I would be very uncomfortable with sending my kids to a class that an instructor refused to let me watch. The instructor should be understanding to the situation and make an exception for you to sit in at least once. It seems very harsh to tell a little child that if they leave again they shouldn't bother coming back. I would pull her out if it were me, but I do understand you wanting her to stick with it.
I think you may be better off letting her stay home a while if that is possible. Pushing her harder to be independant at this stage will most likely backfire and send her a mixed message. With my daughter, I held her more (belly and all) and let her be my shadow as long as she needed to so that she felt safe and secure.
Its very hard to give you advice with having such limited knowledge of you situation, and every child is so different.
It will all work out! Your daughter will calm down once life settles down and the baby comes. Good luck!