I don't agree that she needs the social interaction. We often push kids and over inflate the importance of socialization.
It sounds like your daughter is over stimulated - very much so. This is an extremely stressful situation for her and it is not working.
Not everyone likes, needs or wants that much stimulation from others.
Bring her home and give her space, see what she does when she has a few weeks to relax.
Friends come when they do. Sanity and inner peace, a chance to express without pressure, this is what is precious.
Take the pressure off of her - this can cause long term problems and teach her that her own inner peace is to be ignored - for what? She'll make friends when the time and people are right. If she is happy and has peace she will find the right people and her place in society.
Mom's should listen to our hearts and throw away these "rules" about how to raise a child. Your inner compass tells you. Expert ideas should never override your heart and her signals to you.
She is 3 = don't talk to her like an adult. Lovingly observe her and create a loving and peaceful space for her to discover her life and explore in.
Take it easy on yourself too, sounds like you are working too hard to do it right. Each kid is their own person and has to have time and space to find that joy within.
Less talk, less stimulation, more loving interaction, walks and gentle exploration of the world.
If she desires socialization, find a play group or take her to the park. It sounds like she is just too young for this.
Enjoy her!
N. Marie
www.naomimarie.com