D.G.
Hashimoto's runs in my family, and it is possible to lose weight on thyroid meds. We do better with our TSH optimized to where we feel best. (Most people without thyroid problems have a TSH between 1-2 mIU/L, and there is increasing evidence that a "normal" TSH is below 2.5 mIU/L).
We also do best on diets that cut back on refined carbs (basically white flour white sugar and white potatoes) like low glycemic index or South Beach or Sugar Busters or even Hawaii Diet. And of course with a regular excercise schedule, even if it is just walking or yoga.
Unfortunately you can gain weight when hyperthyroid and hypothyroid. A good rule of thumb is to get your thyroid rechecked if you lose (or gain) 10% of your body weight.
Good luck! Take it easy on yourself 'til you get your levels straightened out.