I had this issue with my child, or at least similar. it was hard to fathom, for me, anyway, if I was MORE firm with her, she would soften, but that is what occurred. I was at a point where I would respond to her screaming and rudeness by backing away and giving in, more and more, thinking she would "come around" or at least stop hollering. ...
It literally took only 1 solid consequence to get her to get on a much better path, and it's been mostly uphill since then. I had to sit in her room for about 1.5 hours listening to her scream and not 'give in'.
This was us. We are both empathetic and sensitive, but our parents were strict and unreasonable in our memories. So we decided to parent in a more empathetic way. Turns out we got it backwards. The reason that we turned out the way we did was because of our parents' firmness. If we are firm with our DD then we have a much more thoughtful and nice DD. If we back off, she turns into a proto-super villain (highly intelligent, manipulative, selfish, mean). When we lay down the law, she's both better behaved AND happier.
Seems we owe our parents an apology (at least for that one, there are other things they did that we are NOT emulating -- like my parents were very, very overprotective and it didn't help me at all when I became an adult).