Recurrent sinus infections, tinnitus, ear infections and dizziness are all symptoms that may be related to being hypothyroid. In addition your doctor has told you your thyroid levels were low. Your latest lab results were too low. Why in the world are you still resisting taking thyroid medication? Why are you wondering about Lyme disease when you are having all those hypothyroid symptoms, and also have low thyroid levels?
A good thyroid doctor will treat a hypothyroid patient clinically, by testing and adjusting Free T4 and Free T3 as needed to relieve symptoms, without being constrained by resultant TSH levels. Symptom relief should be all important, not just test results. Were you tested for Free T4 and Free T3, or Total T4 and Total T3? Before discussing further, please post test results and reference ranges shown on the lab report.