3 YR OLD SEP ANXIETY

My three-year-old daughter is bright, articulate, and creative.  She has never been in a daycare setting and will be starting school when she turns 4.  She is cautious around people she doesn't know and has always taken a long time to warm up to new situations.

I have enrolled her in a dance class and music class for preschoolers where she is supposed to attend by herself.  I am keen on fostering her independence since she has only ever been at home with me.  She attended the first two weeks of each class with no difficulty whatsoever.  Since then, she has refused to go into the classroom without me, and once I'm in there, she refuses to participate. She ends up watching the class from my lap.  

I have continued to take her to these classes hoping that she would warm up to the idea of going in on her own, but she seems to be getting more and more upset every week.  

How do I know if she is having anxiety about separating or if she just doesn't like the class (out of boredom or some other reason)?  She says she's scared but can't elaborate on what is scary.
Is this something she will grow out of in time for school next year?
Should I continue to encourage her to go to these classes and sign her up for different ones?
Which is the better method - going into the class with her (even though she only watches and it disrupts the class)? or taking her home when she won't participate (which seems like quitting to me)?

Could use some suggestions!
Thanks.