Sorry to jump into this thread so many weeks later, but wanted to provide you a link to a bunch of information I have found very useful as I start to wade into potentially having hypothyroidism.
http://thehealthyskeptic.org/...ns/thyroid-disorders This guy falls into the "you have an auto-immune disease" camp, and thinks that in a lot of cases that dietary changes can make a huge impact. But he's also not against the thyroid drugs, and has several well researched posts (all his are very well researched and link to a lot of studies) on what types of the medications will work best for you given your various test results.
He's big into gluten is the enemy, so eliminate wheat and other major offenders like that. Vitamin D would a useful supplement for you, especially if you have hypothyroid and are a lady. This is probably also related to those reporting they need more medication in the winter, it could be the lower sunlight exposure and lower Vitamin D levels.
I'd recommend that you take an hour or three to read over a bunch of his posts and see how it strikes you.
My GP was not a fan of dietary changes, she said pretty much flat out "That won't help." Which I didn't like. I haven't sought a second opinion yet as I'm heading for a 3rd round of tests for my levels soon, and had an ultrasound recently.