Ya know, I think ALL seaweeds are suspect, so I wouldn't rely on them heavily. I know that all sea veggies contain bromide so that's definitely not good.
Supporting the adrenals, is a must. If you are very low cortisol, then rx hydrocortisone like cortef makes since. It'll make the difference in getting well sooner or taking longer to get well. You can also try things like licorice root, isocort and even hydrocortisone 1% cream (apply topically).
Iodoral, which is the tablet form of Lugols iodine, is 25mg of Iodine per tablet. You can get 2 oz of 2% Lugols from swanson's vitamin for $10. Iodoral is around $30 for 190 tablets. There's around 1-2mg Iodine per drop of 2% lugols. There is a yahoo naturalthyroidhormoneadrenals user group that can tell you how to support your adrenals. Besides the Iodine forums on curezone, there is one on yahoo too. It is run by a woman who recovered from thyroid cancer with the help of iodine. Since she had her thyroid removed, she's on thyroid supplement, currently nature-throid.
You should have your RT3 values tested to see where they are. If it is too high, then it means your T4 hormone is not getting converted to T3 but instead to RT3. Taking a T3 supplement, like cytomel, will bring it down.
I would post to Val's NaturalThryoidHormoneADRENAL board on yahoo your latest labs with ranges, 24hr cortisol saliva test would be good if you haven't done it yet. You might need to get your aldosterone and renin levles tested too. Val and her mods will be able to help you put together a treatment plan in conjunction with your doc. Val's adrenals were really tanked due to pesticide exposure. She's recovering, but still needs cortisone because her adrenals just aren't functioning optimally. But because she's on cortef, she's providing her body what the adrenals are supposed to and she's doing well. So there's hope. Ya just need to talk with those folks.