ah..yes...this is my oldest!....Whenever you leave....just leave, do not "egg" it on as my husband would say. Do a quick goodby and then just leave OR when it did get really bad I would leave when she was in another room. Your daughter is going to have to learn that going into hysterics is not going to keep you there and those watching her should not pamper her....this is a power game....they probably need to place her in a safe spot, like bed or room for her to cry it out .....this is kind of like the same scenario as when our kiddos would cry in bed when it was time to go to sleep when they were babies....of course I ran to mine and rocked them to sleep..but in the end, you have to teach them to self soothe so that they can go to sleep on their own.....Your daughter is going to have to learn to self soothe when you leave to do errands.
For her being less social....get into a mommy playgroup....I started my oldest at 2 years....she has her introverted ways...but she has really come a long way and I do believe being in a playgroup for 5 years has helped tremendously with different playmate, mommy friends as well as going to new places.
I belong to a group in Rancho Cordova through meetup.com. There are tons of great mommy groups around Sacramento on meetup.com.....There are Moms, Stay at Home Moms, Work at Home Moms and Working Moms Playgroups...all you do is enter your zip code and select what group you want to look for and will give you the nearest by you.
You are more than welcome to join mine, we do our activies mostly around our area and outside of it as well....if you are in Elk Grove there are a couple good mom groups that my friend started there....Roseville has some good ones too....I have friends in the Citrus Heights ones..and there are a couple in El Dorado Hills....there are more too...
Good luck and be strong!~
Bests~