heavy legs/dead legs

My mother also has Grave's and went through an adjustment period when she started exercising. Could be something about how your thyroid is responding/not responding to the exercise or perhaps your medication (if any) affects things.

I'd go see a sports doc and have them consult with the doc who's managing your condition. The regular doc will likely see exercise as something that can be sacrificed, whereas a sports doc will help you maintain your lifestyle.

Sounds like training adaptation, honestly. That's a sign that you're training hard enough to improve. If you never get those heavy legs during training, you're not training hard enough to truly improve. It's one of the ironic things about training - the WORSE you feel , generally the better you're getting. The least results come from when you feel like a million bucks on all your workouts. A sure sign you're not going hard enough. 

True overtraining takes weeks of high mileage and is unlikely for someone who's not running every day.