You might find this link useful: A Different Perspective on Hypothyroidism | Mark's Daily Apple
If you are going to take the low-carb route, I'd recommend paleolithic or primal eating. A great place to start would be cutting out all grains (if you're not feeling up to this, rice and quinoa are probably alright to have, but wheat needs to be out for sure). Eventually you could consider cutting legumes and dairy as well but I still eat them in quantity even though I consider myself paleo. The basic foods are whole-food vegetables, fruits, meat, fish, eggs, nuts and seeds. The website at the beginning of this post has more information about the lifestyle.
1) eat real food - more vegetables, moderate meat, moderate fruits, less grains, less sugar, less vegetable oils.
2) exercise - moderate intensity cardio, sprinting, heavy lifting, dedicated stretching and mobility.
3) live - relax, de-stress, meditate.
Disclaimer: I'm not professionally qualified to make any formal recommendations. I've just done my homework and I'm my own guinea pig. All of my data, unless otherwise cited, comes from a sample size of n=1 (me).