You need further testing to get a more complete picture. Many of the symptoms especially fatigue can be a result of or contributed to other deficiencies.
Your B-12 levels are abysmal. Many countries do not even start the bottom of the range at something like 500. And most people seem to have fatigue symptoms when below the level of 900 or so. So you may want to talk with your Dr about that and "pernicious anemia". this is caused by the inability or difficulty in your body having the ability to absorb B-12.
Also Vit D3, Iron and Ferritin need to be tested for also, as any one or more of these being low can affect fatigue and in the case of Ferritin also plays a role in the metabolism of thyroid.
Your Free T4 levels are near rock bottom of the range. It is extremely common for people who are less than 50% of the range to experience hypo (low) thyroid symptoms. Your test show you to be at only 10% of the range. A far cry from being in the middle (50% of the range if not slightly higher).
An important test that you should DEMAND be done in addition to TSH and Free T4 is the FREE T3 test.
Understand that at the cellular level ONLY Free T3 hormone is actually used. It is important to point out that you have to be VERY SPECIFIC to demand FREE T3. As if the notation "free" is not attached, they will test for "total" T3 which is an obsolete test of limited value. Think of it this way. The "total" T3 test will tell you how much gas is in the fuel tank. The Free T3 test will tell you how much gas is actually available for the cars engine to run on. You could care less if you have a full tank of gas, if there is a blockage or some inability to actually get gas to the engine.
Anxiety is a common symptom of being low thyroid. in fat all of your symptoms you list are consistent with being low thyroid. PLUS the fact that your FT4 is at only 10% of the range is further evidence of potential if not likily low thyroid condition.
My recommendation is that you get more testing as stated above before you make any decisions to start thyroid replacement medication. I suspect that will ultimately be what happens. But you and your Dr may want to really know what the values of those other vitamins and ferritin levels are in addition to the Free T3 level is before making that determination. And you may want to bring up those other deficient levels and see what relief in symptoms (if any) you have before you undertake Thyroid replacement.
Thyroid meds are a one way street - once started you will have to take them the rest of your life. I was given levothyroxine 100mcg starting 19 years ago because my TSH was elevated - I had no hypothyroid symptoms. Immediately I developed a whole host of symptoms and have been fighting it every since. This included developing Hashimoto's and a bunch of food allergies - had no food allergies before.
Insure that you are getting enough iodine. One 150mcg pill in the AM and PM, (kelp). Avoid starting any thyroid meds., your FT4 is low which may be caused by insufficient iodine in your diet. Give it 1 week to see if the kelp tablets help.
Levothyroxine is a thyroid REPLACEMENT drug, not a supplement. Let me repeat - levothyroxine replaces normal thyroid function.
Free T4- 0.9 (Range 0.8-1.8)
B12- 240 (range 200-1100)
You require both thyroid hormone pill and Vitamin B12 supplement, use methylcobalamin ones.
Also do check your Vitamin D and ferritin levels to rule out other common deficiencies. All need to be in optimal range and not just contained in normal ranges.
If your doctor is not willing to prescribe you thyroid hormone pill seeing your normal TSH value then find another. Normally if your TSH is normal or low normal then your Free T4 should be at least close to mid range in reference range.