First off is the T4 tests "free" or "total". Also please post the reference ranges for the tests. It makes a huge difference as "total T4" is obsolete and of little value. Second you should demand to be tested for Free T3 as well as Free T4.
Finally TSH is at best a screening test, but anything over a value of 3 should be highly suggestive or hypothyroid (low) thyroid and medication. At least further investigation to check both Free T4 and Free T3 needs to be done.
The fact that you have listed common symptoms of low thyroid, in addition to the TSH over 3, in fact over the top of almost any common TSH range would suggest you are hypo (low) thyroid!. At the very least further and better testing ought to be done. And/or a clinical trial of a starting dose of thyroid medication to see if it makes you feel better is warranted in my opinion. You will have to discuss this with your Dr. If the Dr is unwilling to do anything, it is then time to find another Dr who WILL look into this further.
Ferritin is important for thyroid metabolism. So you may want to start taking iron to help with that. I believe another member here (Gimel) recommends Ferritin to be over 100. But I could be mistaken.
People who are low thyroid also are typically low in Vitamin D3 and Vitamin B-12. B12 should be no less than 700 and most need at or over 900 to feel well. D3 should not be lower than 30 and with data for Covid and immune function it is best to be between 40 and 60.
As mentioned by flyingfool it is important to know whether that was a Total T4 or a Free T4. I suspect it is Free T4 measured in pmol/L Since lab results and calculated ranges vary from lab to lab it is also important to know the reference range listed in the lab report. You should always insist on being tested for both Free T4 and Free T3 every time you go in for tests. Even more important than the lab results is an evaluation for symptoms typical of hypothyroidism. So please tell us which of the following you have.
Increased sensitivity to cold (temp. below 98.6 which is considered normal)
Constipation ( have to use fiber or laxatives)
Dry skin (have to use moisturizer)
Elevated blood cholesterol level
Muscle aches, tenderness and stiffness
Pain, stiffness or swelling in your joints
Heavier than normal or irregular menstrual periods
Slowed heart rate
Enlarged thyroid gland (goiter)
Also where are you located? That may make a difference in how you are diagnosed and treated?
Flying fool also mentioned Vitamin D and ferritin. If you will click on my name and then scroll down to my Journal, you will see an Overview of a full paper on Diagnosis and Treatment of Hypothyroidism: A Patient's Perspective. If you will read the Overview it will explain a lot. Also in the material there are references that recommend that Vitamin D should be at least 50 and ferritin should be at least 100. So you need to supplement as needed to reach those levels, in addition to working on your thyroid issues.