First, there is no point in changing doctors, as most doctors know very little about thalassemia. A hematologist might know more, but it's not a given.
HbH constant spring is a more severe type of alpha thal and sometimes will even require transfusions if the Hb slips enough.
Folate is the main treatment. I would suggest getting L-methylfolate and taking 2-5 mg daily. A sublingual B12 tablet can also help. This helps build healthy red blood cells. I would suggest never taking iron as long as you follow a nutritious diet. The iron numbers appear low if alpha thals but this does not mean iron deficiency is present, and as you have seen, iron can cause problems, especially when taken long term. With HbH constant spring, your Hb is going to be low enough that your body is already loading more iron than it needs, so supplements should be off the table. It will only lead to long term health problems and can even cause organ damage.
The children should be tested, because they are going to be carriers of alpha thal and need to know their status before having children of their own. They need to know what they carry and any prospective partners should also be tested before deciding on children.