Hi J.,
I'm chiming in a bit late, but I wanted to share our experiences with sort of the same issue.
My daughters are both shy. My oldest still is, and she's 17. I didn't fully realize the seriousness of the situation with my oldest, and except for her being in daycare when I worked, and then her going to school, she didn't really have any structured and planned activities with children other than her brother who is 17 mos. younger than she is. So, she's never fully overcome her shyness. I blame myself because I didn't realize that I could help her with it. I thought she would outgrow it on her own. I'm hoping she does better when she goes to college, as she *can* come out of her shell, she just doesn't do it regularly or with ease.
My son, I called him my little politician. He would, and does, talk to anyone. He has no problems with shyness whatsoever, and in fact, I think he's a little too.....brave.
My youngest, was shy also, to the point where she would grab my leg and cry, like your daughter does. She was, and still is to some degree, a mama's girl. What I did with her, was to introduce her to as many situations as I could, where I could be involved with her outside interactions. We enrolled in Kinergym at the Y for 2 whole years, and she did preschool for two years, and in first grade, started taking swimming lessons, and is now going into 4th grade and has advanced 3 levels.
My point is that not only only children are shy(although she is 7 years younger than my 2nd youngest, so she seemed like an only child alot). 2 of my children have been shy, and there are ways you can help them to first cope, and then deal with and overcome, their shyness.
My 9 year old can now talk to people without hiding behind me(she stopped that at about age 4, and quit waiting for me to tell her it was okay to talk to other people around age 5), and now will talk to almost everyone(not strangers, of course unless she has permission).
Find a playgroup you can go to with her, Mommy and me gym class, co-op preschool(so you can be there too), library and bookstore story times, playgrounds, martial arts lessons(not just for self-defense, but more for self-esteem and so she can learn to focus in a positive way on herself), and like another mom said, try letting her spend alone time with people you trust, so that she realizes that you don't have to be there all the time for her to be OK.
Best of luck to you both. It will take time, but it's normal, and doable to overcome.
K. W