Hi N.,
I feel your pain. I went through the exact same thing. I would have friends tell me to stop giving in and let them babysit - well, after a half an hour I'd get a call - and they didn't want to babysit again either. I couldn't go to the bathroom or take a shower without a crying fit.
I decided early on that if she is truely that frightened by my leaving, I would just give in to it, and that she would outgrow it at her own time. So, she went everywhere with us - we really didn't have a social life until she was a little over two years old. In the meantime, I had met some wonderful moms/friends in a Mommy & Me class and we held weekly get togethers at each others' homes. As my daughter got accustomed to being in the other homes and being around the other kids, it made it easier to swap babysitting. It was easier with other kids around.
Preschool was another tough separation. She cried everytime I left her for the first year. By the second year she was okay being left at preschool.
In Kindergarten, first, and the beginning of second grade, she would cry if I did not walk her to class. She was fine once she was in class. I had to be there where she ould see me when she got out of school or she would cry.
All of a sudden, she wants to be dropped off - no more being walked to class and is eager to go to her friends' houses to play.
On the positive side of this clinginess - when we were in restaurants or out shopping - she would never leave my side so I never had to watch out for her running off. So I did get to sit and enjoy my lunch/coffee while the other moms were off chasing their toddlers.
I hope this helps you - it is really tough.