Just want to say that I have a 3yo and she's very similar. Delightful sometimes and downright frustrating at others.
I don't like the kind of parent I am at times either, but I think it helps when my 'well' is not empty... KWIM? I think how I respond to a situation is very much about me rather than her. She is the same mischievous little bugger day to day, some days I chuckle and think, 'golly, what a minx!' others I tear my hair out and wish she was someone different So I really need to make sure I get time to replenish whatever it is that I need to keep me positive. Often I need to demand that (from DH). It pays off.
I struggle with her interactions with her sister - she is too overzealous most of the time, and this is my particular battle. I tend not to worry about saying hello and goodbye (even though it's embarrassing when they don't, IMO at three, social situations like that can still be a little beyond their grasp... saying 'goodbye' means that something fun is going to end. Kind of understandable to refuse to assent to that!)
I don't have a failsafe method. Bribery usually works, or distraction, withdrawing attention. Not particularly 'gentle' methods but they are all I have in my limited toolbox.
I am really hoping it's a phase. I am trying to enjoy the good stuff as much as I can.
I hope you find some sort of resolution soon. xx