Liver tastes great if you prepare it right! but to have it be that low there must be something else going on. Are all your other tests normal? Kidney and such?
Low hemoglobin isn't always low iron. It can be low folic acid too. Your doctor should have given you a prescription for folic acid. If they didn't, get a new doctor. Eating half a grapefruit every day is a great way to get folic acid. You can use the grapefruit juice in your salad dressing too.
A great and overlooked source of iron is whole grain cereals. They have to be real whole grain, not flakes or any other processed shape. Quinoa, Bob's Red mill, etc.
It's perfectly normal for TSH to fluctuate. Everything in the body does to a certain extent. Stop taking iodine. Overdosing is even worse than not getting enough. You really don't need all that much. With those thyroid readings I wouldn't worry about Iodine deficiency at all.