For some reason, I don’t think your interpretation:
"You are the meanest mum! I hate you, I want to go after that dog! Why won't you let me go after that dog? You are so mean! You are a meanie!"
is quite what Amber is thinking.
Funny as heck, but you're being far too hard on yourself and personalizing it as a direct verbal assault on you.
Barking and whining are a way that dogs vocalize excitement and are able to release some of that excited energy.
I;d look at more along the lines of:
Okay I'm sitting but, but, but...there's a dog over there and other dogs are so exciting. They get me really excited, see how excited I am. Don't you see him, don't you see him. He's right there. Let's go. Hurry, hurry before he goes away. Come on, let's go, let's go.
Not really, but it really has nothing to do with you. It's more about her energy level when she sees other dogs.
If I remember correctly (and correct me if I'm wrong because I’m just going off a memory here), you initially had ideas of putting her into some kind of doggy day care. Has she had a lot of exposure to playing with other dogs? Does she still play with other dogs?
Aside from her nemesis (and sorry, I don't recall what happened there) how is she with other dogs? I’m thinking she’s the kind that wants to initiate excited play right away.
Personally, I think Bob gave you the perfect advice and so succinctly:
If your using markers to reward the dog ignoring the distraction, just extend the time you wait till the dog quiets down. That may take a bit since the dog could now be looking at the "yelling" as the thing your rewarding if the timing is off.
Once she’s sitting with focus on you, her attention can start to be expanded but she needs to learn that her tantrums have absolutely no effect on you and will get her nothing.
I see you said you "usually" use a toy. How are you delivering it? She sits, you "give" it to her, then you take it back, then she starts to bark and you give it back to her?
I know it's probably a little tough as it migbht be somewhat embarrassing or distracting to you but this is where it's getting reinforced for her. She's getting her toy and it's getting you moving.
Her vocalizing should not get her one word or movement out of you unless you've actually trained a "leave it" or "quiet" type command.
At this point, she's not listening anyway. Any words coming from you are just adding to the excitement. Quite, calm is what needs to get your attention, bringing on the great rewards and starting the fun times, with you.
And don't underestimate what can be fun for a dog. "Let’s see how frustrated we can get our human today", is a game I'm sure they teach each other. Don't play - trust me, you’ll lose.
As Bob mentioned, this may take a bit because her tantrums have been reinforced. You’re going to have to sit her out.
I'd probably be working on a "Leave it" with her and practicing my ignoring skills whenever she gets into this state. Does she do it around the house at all? If you're finding the ignoring difficult, you can always turn your back to her. Stand as still as possible and don't say a word until she settles and stops. Then "Yes" and out with the toy or treats or whatever she finds rewarding. If she starts again after the reward, turn your back again.
Once she starts to get the idea and is calming quickly, you can name it or put it on a cue. I'd want at least 20 good responses with something like this before you name it though otherwise, again, your words are just adding confusion to the already escalated mix.
And Jodi, for what it's worth, I think you're a very good mom. She simply doesn't know any different way to act right now. A mean mom wouldn't be teaching her the manners she needs to get along in this world. You can do this.