an update...I gave her the one carrot and when she finished eating it and nuzzled me for more I said NO loudly and walked away. She immediately circled and did a body block.
The mare's reaction shows that she doesn't respect you AND she doesn't have a working relationship with you. The carrots are spoiling her and the stick is frightening and challenging her.
Horses react VERY QUICKLY. Well broken horses react quickly only to their training, such as quickly taking cues for walk, trot and canter and movements in a particular discipline. Otherwise, they are calm AND obedient, and this is what you want to achieve, NOT getting her to jump quickly away from you.
We continue to train and correct the little disobediences. I have trained my 16'3hh, 6yo gelding to put his head over the stall gate and he holds it there until I halter him. I walk him out and we "halt" at least 2x on the way out. Often I walk him through the 32" human door, then back him a few steps, especially if he isn't paying full attention to me. He knows "around", and he walks around the gate, I shut the gate, he puts his head over the gate with the lead hanging, and I take my time unhaltering him. Often, he wants me to scratch his face, and, since he's not using me as a scratching post, I do this ONLY at this time. Anyone handling my gelding has no trouble with my 1,400 lb horse bc he doesn't drag you out of the barn. THIS is the training that makes a horse usable. YOUR mare would NOT behave like this.
Aggressive behavior will make a dog
submit. Aggressive behavior will challenge a horse. If you horse wants to get into a fight with you the horse will win.
Watch Intl. x-country, where the jumps, like at 2012 Aachen, are only 4 ft. wide (4 ft. high) are they are jumped coming OUT of the water. THIS is the obedience that we all train for, and it takes years to accomplish.
You are in WAY over your head, and fixing to get hurt. There are many, many, trained horses out there with owners who have lost their jobs and cannot keep them any longer. Get your ego out of the way, and dump this problem horse. Find a better animal, or don't be a horse owner until you invest $ in lessons, so you know what a horse's training is supposed to be.