What your doctor is looking at is the elavated TSH lab test over the max of 3.0 - clinically seeing you may be a bit hypothyroid. He is apparently trying you out on a thyroid replacement hormone to see if you respond to ridding your symptoms and attempting to bring that TSH lab test down into a more normal state as a number figure goes.
The Free T3 test is important as stated - the T4 lab you recieved is useless. Total T4 lab measures the T4 hormone inbound in the system that will go nowhere to sabilze you at all.
Give the med a try - be prepared you will not have any accurate testing for at least 6 weeks after starting - Track your symptoms daily to see if you improve or decline. Basal daily temp checks are good too. Compare those daily trackers to the tests you just had done and the future tests you will recieve.
Do yourself a big favor though - ask - for the seperate TSH ( not reflex) the Free T3 and the Free T4 tests at your next blood draw.
I would also advise some antibody testing. I too have hashimotos/ and am hypothyroid/ plus have rhemotoid arthritis.
Test results and symptoms are hypothyroid. Med. will be helpful. When you go back for next appt. suggest that free T3 also be checked, since this is the most active component and largely regulates metabolism. If you are interested you can confirm the hypo t diagnosis by checking your temperature. Here is a good link for that.
http://www.drrind.com/tempgraph.asp
It takes time. I am on synthroid 50mg. On and off different brands since june. Take one day at a time, oh so hard to say. lol. Watch for symptoms like can't sleep, ot sleeping to much. Your body is going to be stressed out for awhile because of the changes. This forum helps with that too. I went through different phases, but everyone is differeent, anxiety depression was one of them. Just keep us updated, try and take it the same time of day on an empty stomach. Any other meds at least one hour later. You should eat at least 30 min after meds. Good luck. Its a long battle will be worth it in the end.