My heart goes out to you, as this is my life too. When my daughter was 2, I also asked if this was my fault, what was I doing wrong, what was wrong with HER? I won't say "I know just what you're going through", because I don't. Each child is so unique, and there are no "one-size-fits-all" solutions. I will also say that the people who brush your problem off with a "she'll grow out of it" are exactly the people you should ignore. Your daughter needs to learn coping skills, and she needs your help. She also needs your endless patience, and that is going to be so hard for you. As she gets older, that clinginess is going to be more embarrassing. So, start working now! Here's what I did, am doing, and will continue to do:
First, I raided the library. I found a few books to be VERY helpful: The Difficult Child by Stanley Turecki and The Out of Sync Child by Carol Kranowitz. (The Difficult Child is a horrible name for such an excellent book - don't be put off by the published date.) There are others, many many others, and you can look for yourself to see if the authors have written a book for YOUR child. You can peek at some of the child anxiety books too, but they are mostly geared to school-age kids, and not as practical for preschoolers.
Next, DO THE EXCERSIZES in the books you do choose. Reading is nice, but writing things down, talking with your husband, making charts, schedules, it all makes a difference. Personally, after I wrote down all my daughter's frustrating behaviors, I realized only TWO of them were things I could help her change. The rest of her behaviors were part of her personality, and I wouldn't change that for ANYTHING. A clingy child is a challenge, but all the hugs and kisses that go with it are a blessing.
Give your daughter words. You will probably find that some of the shyness wears off over the next year, as she is able to identify her emotions correctly. Make sure you use the right word for each situation: angry, sad, frustrated, scared, confused, etc. Words have power and power inspires confidence.
Keep doing the activities. Constant exposure to the bright and messy world of childhood will held desensitize her to it, and hopefully someday she'll be ready to participate. In the meantime, YOU participate. You can't force her to have fun, but you can show her that you are having fun. However, you may want to limit yourself to small intimate classes, and focus on just 1 or 2 close friends for playdates. Listen to her preferences too. Knowing she has a say may (*may*) make her more vested in getting involved. Or maybe not.
Stop worrying about what the other parents think. THEY HAVE NO IDEA HOW HARD YOU ARE WORKING. Your focus is your child. Do what your child needs, when she needs it. Her sense of self-worth will grow, and she WILL someday let go of your hand at the playground.
And TAKE A BREAK. It's exhausting. Get out of the house by yourself every week (even if it's just an hour). It's good for your daughter to spend one-on-one time with her Dad or grandparents. And it's vital that you get some private time.
My daughter will be 4 soon. She's in preschool now, and she can sit by herself for library storytime (but I still have to stay in the room). At the playground, I'm the mom climbing through the tunnels and squeezing myself down the slide with a "big kid" on my lap. After about 1 1/2 hours of that, my daughter will go down the slide by herself. I see that as a HUGE accomplishment, and I am getting very good at ignoring any stares from other parents. I lavish praise on her for the SMALLEST accomplishment (good job saying "hello"!) and she gets treats and presents for bigger stuff (you COLORED at school today? here's a lollipop).
Sorry this is so long - but one more thing. I enrolled her in a co-op preschool, where the parents take turns helping out in the classroom. She started after she turned 3, and I am seeing amazing progress on the shy scale. If you can afford it, I definitely recommend it.
I hope you can find the patience to work through this challenge. It does get easier, but for some kids, it never really goes away. Good luck.
-K.