My oldest little girl (the second child) is 5 1/2 (turned 5 in September), and she is also incredibly shy. I have had her going to a Mother's Day Out (just 9-2:30) since she was 1 (1 day a week when she was 1 and 2, 2 days a week when she was 3 and 4, and now 3 days a week when she's 5, but we still had attachment issues when she started back to school this past fall. For a good month and a half, close to two months, she cried every time I left her at school, Sunday School, or dance class, but after that transition time (which seemed like a long transition time to me!), she is actually smiling when I leave her at all those places. She's still shy at new places and around new people, though; we can go to a birthday party with kids she knows, and she'll cling to my leg for close to an hour before she'll warm up to play with the other kids.
I'm telling you all of this so you'll know that I know where you're coming from. But here's the thing. Even if you leave your daughter screaming and have to have her pried away from you, within minutes of you leaving (usually), she'll be happily playing with others. I know how hard it is to leave your little one crying his or her eyes out, but it's really harder on you than it is on the kiddo. And by giving in to her screaming by not taking her back to the places where she screams when you leave her, you're teaching her that that's all she has to do to keep you with her.
Since you are worried about the socialization of your daughter as well, I would recommend putting her in a Mother's Day Out part-time, a couple of days a week (the kids who go two or three days a week usually have less trouble adjusting than the kids who go just one day a week), or if you don't want to do that, try to have a regularly scheduled playdate once a week with some friends with kids the same age.
Something my daughter's teacher this year suggested to help in the transition period was for me to leave a picture of me with her so that when she's missing me she can look at it, and that really helped her. It was a big deal with she finally announced that she didn't need the picture anymore!
I do have to admit that after years of this, I've become rather inured to my daughter's crying. She would cry at places where I know she has fun and wants to go (like dance), for no other reason than that "she wants me" (I hear that chant A LOT!) While it's still hard to leave her crying, I also don't give her a choice about where she's going, and she gets over it. (Though it really helps if the person in charge where I'm leaving her will get her to do the activities rather than just letting her sit there and sulk.) I'm really not callous about it (well, not too much!)--when I take her to school, I still stick around for the hand-washing routine and at least 4 hugs before I'm able to walk out the door, but it feels so much better to leave her with a smile rather than tears.
Finally, I will leave you with a thought that someone shared with me when my daughter was crying everywhere we went this past fall: she's a child who will always want to come back home for the holidays and visits. When I'm frustrated and worn out by my daughter's crying, I remind myself of that, and it's some comfort to think of a positive outcome to her attachment to me.