This would be unusual for thyroid meds. A neurologist may help identify the cause.
Don't know that it's caused by the meds. I had one headache like that before I got the thyroid diagnosis. The headache lasted 3 months, 24/7. Yep, pressure is the word to describe it.
Now it will only return when the meds need adjusting. I could be headache free going to bed, but when my head touches the pillow the pressure builds.