It isn't IMPOSSIBLE to lose weight without exercising but the exercising sure makes it a lot easier. Unfortunately, an under active thyroid is going to complicate things too. I am sure your doctor has explained a lot to you about how the thyroid operates and why it is so necessary for it to be regulated. You may need to give it a little more time, for it to be fully regulated and you may need to talk to your doctor to see if the artificial thyroid levels need to be increased just a tiny bit. I have been on Levothyroxine for about 15 years now, due to Graves Disease. I can tell you that I started gaining weight about that time, now I can't blame it all on my thyroid, I fully admit that I over eat and under exercise...a dangerous combination!!!
Unfortunately I also think that our metabolism changes after we start having children...I used to be able to eat just about anything I wanted and not gain weight...those days are long gone.
I dont know how old your children are, but maybe you could incorporate some "exercise" into play with them...it would be beneficial for all three of you. Go to a park, take a walk, find some fun active games to play. It will help you with your weight loss goals and start your children on a healthy lifestyle.
Start a food journal...keep track of everything you eat for a week or two...(there are good calorie counting sites online....look at Calorie King or Spark People), you would might be surprised at how many calories are in otherwise healthy food. Dont think of this is a foot race or a sprint...think of it as a marathon...you need to develop a new way of thinking about food and exercise...and carry it with you for the rest of your life. (Easier said than done...I KNOW what I should be doing but that doesnt make me DO it!! I have lost...and then regained about 50 lbs in the past 2 or 3 years!!!)
Good luck honey!!!