Being hypothyroid is definitely a real drag...it's so hard to muster the energy for exercise, but I've found that the more I exercise, the easier it gets and the better I feel.
Getting on the proper dose of meds is key. It can take a little while to get that dose right, but in the meantime, you can take control and start making lifestyle changes with your diet and your exercise. And even if those changes don't show up on the scale right away, you'll be on track to a healthier you!
One trick I learned that helped me when I was very discouraged and frustrated was to count just about any extra movement as exercise-- literally something as small as parking at the other end of the parking lot at work. The mental lift from walking an extra 50 feet usually spurred me to do more, (oooh, now I'll take the stairs, now I'll skip the donut, now I'll walk on my lunch break) even if it was in bite-sized bits. And if I screw up (usually involving said donut), I don't dwell on it much or get my binge on-- I just apply the brakes and try to do better next time.
I think we beat ourselves up mentally. Sometimes showing yourself some kindness and building on positive feelings instead of dwelling on negative ones is enough to keep you going, one little bit at a time.