I, too, was recently diagnosed with Hashimoto's thyroiditis (January 2012). I have had to learn to slow down a little. Luckily, I have small children so we've always tried to do afternoon naps. If we don't nap, we get back to the room a little earlier (for my sake and theirs). I have never picked up a Guest Assistance Card; however, they are easy to obtain and highly recommended if you need one. I put together a list of air-conditioned attractions when I traveled to WDW while pregnant. Just PM me and I'll e-mail it to you.
While at WDW just three days after my diagnosis, I had a "spell". I just had to accept that I needed to stay in the room (in bed) for an afternoon. Don't push yourself too much (to the point of exhaustion). It may take more time to recover than it would to just take a break when you need it.
Cheryl is right about places to sit & rest. If you cannot find one in a shady place, go into a counter service restaurant and have a seat. It's perfectly ok - no one will kick you out.
Above all else - Have Fun!