Find a new Dr ASAP!
You have all classic symptoms of Hypo and your FT4 is BELOW range! Clearly your pituitary isn't doing its job to pump out TSH in order to get your thyroid to produce, and/or your thyroid is simply not working. But i would tend to put the odds on the former.
You must also consider getting vitamin B12 tested, as being deficient in B12 causes severe fatigue and also tingling in the extremities. Both of which are symptoms you report having.
In addition you want to test for iron, Ferritin and Vitamin D3 in addition to the B12. I would also recommend the two different antibody tests for Hashimotos which are TPOab and TGab.
Having a Dr who seems to dismiss symptoms so severe that it causes a person to stop working a career is in my opinion bordering on malpractice. Especially given your symptoms and blood labs so clearly point to hypothyroidism!
As long as your Dr only looks at TSH, they will keep you feeling horrible.
With those symptoms and labs, you shouldn't really need further "evidence" of what your problem is; your doctor would be able to see it if she were a very good doctor, but many doctors miss Central/Secondary hypothyroidism because they are too focused on the TSH and since it's, basically, "pristine", they think there's nothing wrong, in spite of the low thyroid hormone levels.
You can try to discuss with her the concept of Central/Secondary hypothyroidism and see if you can get her to, either, give you a trial of thyroid medication or a referral to an endocrinologist. Unfortunately, even many endos miss Central hypothyroidism because they, too are concentrating only on the TSH.
Antibody testing might come back negative anyway because, with Central hypothyroidism, the problem is with the pituitary gland, not the thyroid.
Your profile says you're male... is that correct?
Have you considered changing to a different doctor?
Yes, you should be concerned and either ask for a referral to an endo or find a different doctor asap. Your doctor is totally ignoring the fact that your Free T4 level is completely out of range... That means your thyroid is not working.
I can almost guarantee that all he's looking at is that lovely TSH sitting at 1.6 which is where every doctor wants every TSH level to be... Shame on him!!
Of course, with your Free T4 and Free T3 as low as yours, we'd expect TSH to be higher than yours since TSH is counterintuitive and typically, rises as thyroid levels decrease. In your case, we'd suspect Secondary or Central hypothyroidism, which is when the thyroid works, but there's a problem in the pituitary/hypothalamus axis in which there isn't enough TSH to adequately stimulate the thyroid.
I don't even have to ask, but I will... what hypothyroid symptoms are you having?
Have you ever had antibody testing done to determine if you have Hashimoto's? Have you ever had a thyroid ultrasound?