Hi I have both, the diseases often travel together over time.
Symptoms of Hashimotos include:
slowed reflexes, low body temperature, slow heartbeat, irregular heartbeat, blood pressure irregularities, loss of outer edge of eyebrow hair, coarse, brittle, strawlike hair, loss of scalp, underarm and other body hair, dry skin, dry mucus membranesjaundice, paleness of skin./lips, puffiness around eyes, slow movement, slow speech, hoarseness of voice, enlarged neck/goiter, swelling of feet.....in addition can have high cholesterol, cold hands and feet, difficulty losing weight, constipation, fatigue, brain fog, pains in joints..particularly the hands and feet, depression. low sex drive,
eye problems....dry and gritty/achy eyes, eyes that are sensitive to light, tinnitus ie ringing in the ears and/or hearing loss, more infections, allergies,sleep apnea, lightheadedness, chronic headaches, Source "Living Well with Hypothyroidism"
So yes you can see that your symptoms could be related to the hashimotos....it is not always the case that the T3/T4 levels as "normal" actually are at a level where the person feels well. It is very individualistic. You might try to doctor again and see if a small dose of synthroid..helps you feel better.
Check out the resourses...ie books from your library to learn more.