I noticed that you have a fairly new baby. My guess is that some of the behavior may be connected to her trying to get attention. It sounds from your post that you are a very attentive and caring parent, but in the world of a two-year-old, any attention diverted away from them is too much. They literally want it all. She knows that she has found a way to get your attention and keep it, so she keeps repeating the attention seeking behavior. Kids do what works and what gets results. Generally, when you try to get them to change their behavior, they up the ante and the behavior gets worse before it gets better. My suggestion to you is to drop her off,tell her you love her and you hope she has a great day, give her a hug and kiss and leave. Don't let her see that you are distressed in any way. Make sure you do this consistantly for an extended period of time. It is easy to give up after a few times and say, "Oh, this isn't working." But, with behavior modification in children, something can literally take 20-30 tries before the behavior is modified. This means it may take a month or two before she sees that her behavior is getting her no where and stops it. As I stated before, she may even scream louder and throw more of a fit in the hopes that she will get you to stick her around. But, show her it doesn't work and she will eventually stop. If you give in or change your reaction even one time, you will undo all of the work you did and will literally have to start again from square one becasue in her mind she will see that she kept the behavior up for long enough and it worked!!!! The key is consistancy.
Remember, teaching your child to be independent and functional in the world isn't mean. It is the best thing you can do in the world for them. Let go of your guilt and just feel confident that you are teaching her to function as a strong, confident, independent child.
Good luck!!!