How to discipline a strong willed 3 year old

We’ve had quite the “phase” in our house lately. For two weeks the Princess has been a holy terror. She has been in trouble so often at preschool, I’m afraid they are going to kick us out. She refuses to nap, or even sit quietly on her mat so the others can sleep. When we take things away, she says “I don’t care” or “That’s okay”. (We call that Sassy Mouth around here and she’s got it down PAT unfortunately.) She’s been in time out, she’s missed her ‘fun activities’ (gymnastics, music class, going to McD’s to play-don’t judge me- they have an indoor play place people and it’s freaking HUMID in Tennessee right now). Nothing is working. My mom passed down the fly swatter she used on me tonight, in the hopes that we can regain some control. I haven’t used it, because while she IS defiant- when you forcefully tell her NO she cries like you’ve broken her heart- but then proceeds to do whatever I told her No about. I’m at a loss. The terrible twos were a breeze compared to this. Who knew the threes would actually be worse? A friend at work recommended Dr Dobson’s book about strong willed children, but I need some help NOW. And I’m a librarian after all, I research and figure stuff out all day long so why is this so dang difficult?!?!? Any ideas of what we should try next?