Kathie, Have they done an ultrasound of your thyroid recently?
Do you mind my asking how high your TSH is? I feel best when mine is between 1 and 2. That's the "new normal" endocrinologists shoot for. It used to be anything above 5 or more (depending upon the lab).
Hashimoto's can be a gradual thing. There can be destruction of the thyroid that can make you feel horrible - especially if you only have a half of one - but the T4 doesn't necessarily change until late in the progression.
Have you had a brain MRI?
I agree with Limpy. I feel better on NatureThroid (it's from Canada). I was on Armour Thyroid too but it was discontinued for too long so I had to switch. My cardiologist said that many of her patients felt much better on those "natural," desiccated thyroid drugs.
Pred will only add to that calcium deficit. They started you on a fairly high dose of Pred. Are they going to wean you down right away? Or go every other day with like 20/10 or whatever?
Doctors have this "protocol" in place where they reach for the Pred before other treatments. It mainly has to do with cost. If you can wean down to a place that still helps your MG but causes less destruction of your body, that would be smart. You don't want yet another gland (adrenals) to tank on you. Of course, you wouldn't want MG tanking you either.
On your CBC, what is your MCV? If it's below normal, that usually reflects iron deficiency anemia. If it's above normal, a B12 deficiency. The problem, however, is if you have both deficiencies present, it can be in the normal range! Yes, they should check your B12 because a deficiency of it can reduce both red and white blood cell counts, though not necessarily make them below normal.
Are you tired? Any numbness or tingling or nerve pain in any extremities? Headaches? Back pain? Any of those symptoms can indicated a B12 deficiency.
Have you been tested for celiac disease? People with CD can have no Gi symptoms but can have deficiencies that can cause all kinds of problems.
The important thing when having any deficiency is to determine the CAUSE of it. For example, I was tired from my B12 deficiency. The deficiency causes the sleepiness but my doctor did not try to figure out the cause of the deficiency!

For me, it wasn't pernicious anemia but achlorhydria. They can do the intrinsic factor antibodies and the parietal cell antibodies to figure out if you have PA. If you have PA, it also means that you have achlorhydria. The antibodies attack the area that produces stomach acid. You need stomach acid to properly digest your food so that it and all of its nutrients can be absorbed in the small intestine. I had a low calcium, albumin and iron too, which are now normal.
Do you take calcium with vitamin D? Don't have anything citrus when you do. Vitamin C helps absorb iron, which you might need too. It's important to space those supplements apart during the day. And it might be a good idea to take vitamin D and calcium twice a day.
I can't do the 70's and up. It quickly tanks my MG. You need to be careful in the summer because the effects are not only immediate but cumulative. Try to find ways to stay cool.
This is rather silly but it works! When I do have to go outside around the house, I put either a cool, wet wrap around my neck or what's called a "Boo Boo Buddy" in each bra cup.

Yup, you read that right. They are these gel packs used for kids - because they are not too cold - and are so useful for cooling down.
Those hand-held fans are great to keep in the car too. I used to go out with a thermos with both water and cool packs in it when I ran errands. My car's AC is fast now though. You have to set yourself up for success in the summer.
Okay, those are my thoughts. I hope you don't get worse. It sounds like you have great doctors and that's half the battle.

Annie