I'm not Dr. Lam, but I've studied neurology for a long time.
You should not experience anxiety if you already have a deficiency in adrenal hormones. The adrenaline production will help you have less anxiety, not more. Hypothyroid means your adrenal glands are NOT producing enough adrenaline and by using tyrosine you will be giving your adrenal glands what they need to make adrenaline.
It's the adrenaline that is the "upper" not the downer. Cortisol is the stress hormone, not so much adrenaline.
Take 500 to 1000 miligrams daily first thing in the morning on an empty stomach. Taking it with any kind of protein (animal or otherwise) will stop it from absorbing properly.
I would recommend getting a book by Julia Ross called The Mood Cure for more information.