In my experience it is helpful to have your antibodies tested, Free T3 and FreeT4 as those are the unbound hormones that your body has available to use. In hypos it's also very common to have low ferritin which is a storage protein for iron and possibly a low B12. I don't know about you, but, if you are gaining weight, feeling crummy the last thing I want to hear is to exercise! I know that docs mean well, but they just don't seem to get the fact that we are lucky to even be dressed some days. That some of us had to take a nap after taking a shower because we were too worn out! I have three kids and know what it's like to be tired due to family life, house and all. There is no tired like hypo. Eat a more protein enriched diet, try to drink pure water, certain vitamins and supplements helped me and so did using natural dessicated thyroid. Hang in there, educate yourself, you are your own best advocate. If you haven't already, try to read a couple of books (if you can concentrate), Thyroid-Guardian of Health by Young and/or Hormones, Health and Happiness by Holtze. These are excellent books with a plethora of information.
A high TSH means you need more thyroid hormone treatment - I would recommend testing more frequently for the next year or so as it seems the thyroid is becoming increasingly underactive. I imagine you have Hashimoto's - an immune system attack against the thyroid which is the most common cause of an underactive thyroid in the US. This can be confirmed by testing thyroid antibodies (TPO and Tg). The immune system is typically not impaired in any other way.
There is no other treatment that will lower TSH. I recommend regular exercise and multivitamin supplementation to all hypothyroid patients. There may be some benefit to B-complex as well as selenium supplementation. Iodine is not necessary unless you live in part of the world that is iodine deficient (the US is not one of these).
Individual symptoms vary and some patients don't feel hypo symptoms until the TSH is severely high (even >40 or more in some cases I have seen). The average replacement dose is 0.7mcg/pound (based roughly on "ideal body weight" - so this may be high if computation is done on actual weight in patients who are significantly over-weight).