C.P.
Sounds to me like she is definitely NOT ready for preschool. I think also that with a new baby sister, she's still just getting used to a lot of changes in her young life and wants your attention. For sure it is exhausting! Have you tried letting the nanny watch the baby and spending some quality alone time with just the 3 year old each week?
After our 12 month old was born last summer, we really had a rough time with our son, then 3. He was a little terror for the nanny... hitting her and basically acting like a big jerk. He was just dying for more attention from mom & dad.
It really sounds like she's longing for your attention and knows she'll get it with this kind of clingy behavior, same way our son knew he'd get it by making us angry...
When you leave her with the nanny, does she cry all day long for you, or just for a few minutes? A few minutes of crying is probably pretty normal... even at 3.
Also, I probably wouldn't give any options about who is going to watch her.... i.e. if she needs to go to the pool, then that's it. Pretend like you have to leave for work or something. Then she's not thinking that she could just stay home with you...