It takes some time to get the proper dosage for you. I have been taking several different strengths of synthroid myself because my MD has not gotten me theraputic yet.
If your labs are off you should be having them checked every 6-8 weeks.
I did schedule an apt with a thyroid specialist not because I am worring of cancer but that I am on in my 30s this is something that I will have all of my life, also I have a very good GP, but I want someone to educate me more about what is going on and I feel that a specialist will be more comfortable answering some of my concerns.
You should ask for thyroid antibody testing if you have not had it. A couple of blood tests. Try to get your bloodwork (TSH/Free T4) every six weeks until you levels look better. Your TSH should between 0.8 and 1.8, give or take.
If your doctor palpates your gland and feels lumps, an ultrasound is a very simple test that can image the thyroid and tell you a lot about it's physical condition.
A good GP can order all of that. If it gets more complicated, then a specialist is in order.