Please give the quoted reference range too as it vary from lab to lab.
Is it really T3 or Free T3?
Anyway it seem like you are moving towards hypothyroid due to the increasing TSH number. It can be due to many reasons especially adrenal problems like too much stress, lack of proper sleep, bad diet especially sugary foods etc.
Anyway the best you can do is make sure you are not deficient in these by doing these blood test.
Vitamin D3 (25-hydroxy D3), Vitamin B12 and Ferritin
If its deficient or not optimal then supplement it to make it optimal. Also change the daily habits if its affecting sleep quality, stress, diet etc. These modifications itself can give a huge relief to many of the hypothyroid symptoms and TSH too can reduce without any medication.
If TSH is still not decreasing after some months then its good to start on low dose thyroid hormone pill. Natural dessicated thyroid or NDT is best approach though most doctors prefer levothyroxine. From my personnel experience levothyroxine is good at initial stages like when starting new but after some months it seem to make situation worse especially due to decreasing T3 numbers as levothyroxine supply only T4. NDT on the other hand is a natural approach and gives good feelings consistently. Anyway the starting dose must be set very low and then increase in steps like every 2 weeks at least till the symptoms get better. Once symptoms get better do the thyroid lab test to confirm the situation.
As des900 requested, please post the reference ranges for the T3 and FT4, since those vary from lab to lab and have to come from your own report for the best comparison.
Also, please verify if the T3 is just T3 or if it's Free T3, since T3 is Total T3, which isn't the same as the active Free T3.
Yes, your vitamin D at 7 is way too low; I'm surprised your doctor didn't start you on a mega dose, which is, typically, 50,000 IU, once/week. At any rate, you should test your vitamin D again in a few months to verify that it's coming up as it should. Optimal Vitamin D is 60-80, so you have a long way to go.
Once we know the reference ranges for the T3 and FT4, we can be more helpful.
semo990... the link to your labs would be okay, but it doesn't work; please post the actual labs and reference ranges for us.