I think there may be something else going on besides thyroid problems, but I don't think it's an adrenal problem. A lot of the stuff that you hear about "adrenal fatigue" and that kind of thing is...bunk basically. There is no medial basis for it.
There does need to be time in between blood tests in order for the new dosage of hormones to balance in your body, otherwise, the tests won't be accurate. Usually minimum of 2 months. If you don't think your GP is doing a good job, you can always go to an endocrinologist. They do have experience with all of the hormones in the body. I would trust that much more than a home kit off the internet. Why would their test be more accurate than a regular lab's?
You say you have breathing problems. Have you had your lungs checked? Are you asthmatic? What are your other symptoms besides breathing issues and balance problems? Are you dizzy?
What specific hypothyroid symptoms do you have and have any of them gotten better since you been on the medication? What was your TSH originally before you went on medication and what was it at your last test? I'd like to see how severe your case was originally.