Symptoms of the disease can be found in this article:
If you suspect your dog may have the disease a full thyroid panel should be done which includes T3, Free T3, T4, Free T4 and for dogs in a breeding program TgAA should be done.
Having full blood work is not enough as it will only give you the T4 reading. If that reading is out of range it is not a definitive diagnosis of thyroid disease and it would be important to do the full panel as well.
I recommend sending your blood to Dr. Jean Dodds in California. She is a thyroid specialist. Her prices are great and it is very inexpensive to ship via the USPS.
Here is a link giving all the instructions for your vet to draw the blood and how to package it properly.
Note in the above link what Dr. Dodds says about the THS blood test for thyroid.