One of my girls went through the same thing. What worked for her was what worked for me when I was a toddler.... don't tell my Mom she was right though!
What I did with my daughter was I started telling her, "I'll be back" EVERY time I went anywhere. This included if I ran a short errand ... went out to get the mail ... even if I just went into another room for a second. Upon returning I would make a big deal .. lots of hugs and kisses and I would tell her, "See? Mommy ALWAYS comes back!"
It got soooo tedious at times and seemed a tad silly ..... but it worked! Within a week, she went from crying hysterically to just crying when I dropped her off somewhere (Which is just the standard child manipulation of Mommy thing! I know all about that from my days teaching pre-k!) and stopping the second I was out of the room. It's also key to stay calm when you take her places .. act like it's no big deal .. the same as when you practiced at home. A quick "Goodbye, I love you and I'll be back!" .. don't linger or let your stress over the situation feed into hers.
I think the process took about a week and a half to really click with her.
If nothing else seems to work, just remember .. it's a phase. It WILL get better in time!