Having thyroid problems can itself cause anxiety, since your thyroid hormones affect you and also affect your adrenals. So if this is a byproduct of your thyroid problems, an antidepressant might help, might not, but it isn't addressing what might be the cause, which is your thyroid. Have you spoken to your endocrinologist about this?
I'm so sorry about your diagnosis. I hope you're doing well now.
Yes, Lexapro can make you feel tired, dizzy and out of it when you first start taking it. This should get better in a few days. Almost all of these meds can have start up side effects but they usually do go away. If you're still feeling really bad in a few days give your Dr a call. 5 mgs is a small dose but your Dr may cut you back to 2.5 mgs and work up more slowly.
I take Lexapro myself and it took about 3 or 4 days before those side effects started getting better when I first started it.