Write the letter. Send it in. DO IT NOW. NO you do not need to put in anything about your pregnancy. Say what i told you to say. Do not say anything more, anything less. DO NOT go in to a bunch of issues.
Especially in your comments to the unemployment system, DO NOT mention, "I was a good employee, sometimes we were off for a few weeks at a time, and I did not file for unemployment." This means you were a foolish employee. Though the jerk might have fired you for filing for unemployment, he would have been sanctioned by the unemployment system for doing so.
I agree with your husband in one sense, You were working for a jerk. It is not uncommon to find jerks operating their own offices, practices, etc. and using the very worst of labor practices. They feel that others should be working for them for free anyway in the right sort of world, and that any request to provide living wages, decent treatment or protection from arbitrary firing is a terrible violation of their rights. Their very tenderest area is the pocketbook, which is where unemployment insurance will eventually hit them.
However, you are such a pushover! "I'm pretty down with it not being approved," you say. Aaaugh!
He liked you okay to begin with because you were probably perceived as a wimp and he could bully you. Even though your performance may not have been perfect, you did your best, you tried to please him, you always did your best to be a good employee. But then ultimately, when you became pregnant, he chose the other person, whom he probably respected more if she was a little more assertive to him. After all, you had already made the mistake of telling him you had to leave town soon anyway.
But turn this appeal letter in. Immediately. Do not put anything else on it except what I said to put. Is that clear enough? Do not over-think this. You will get to tell the whole story when you have the hearing. That will be later.
You will be required to register for work DO NOT bring up that you can't work because you are pregnant. Or that you don't think it is a good time for you to look for a job because you will be relocating with your husband soon. You go in there and state that you are able and available and actively seeking work. Because if you hadn't been fired, you would be working. Pregnancy is not an issue with your receiving benefits unless you yourself put restrictions on your work availablity because of the pregnancy. Keep making your weekly certifications for benefits each week as you have been instructed, even though you are not receiving checks.
That said, it sure does not prohibit you from, (in the HEARING, NOT in the appeal letter) mentioning that you noticed a big difference in how you were being treated after you told him you were pregnant. That's reality. He decided he needed to get rid of one of the people in his office. Since you were pregnant, you were going to leave soon anyway, and she had come back to work, it was you, but if he laid you off, you'd get to draw unemployment insurance for sure. So he had to make up a bunch of reasons, probably talked to someone about exactly how to do it, and fire you.
But you are not going to roll over for this. Though you knew he was a jerk and that he is trying to mess over you, you are going to do the appeal and stand up for yourself, and give it your all to win this. As we keep telling you, you have nothing to lose, a good chance to win.