I would say to stay on the Synthroid, you'll get more answers if the correct tests are done. It won't be completely out of your system in 2 weeks anyway (I've heard 6 weeks to 3 months). You'll know how well you're converting if you see your free T4, free T3, and reverse T3 numbers on synthroid.
I would caution you to make sure that your new doctor is going to request the correct tests and treat your symptoms. You might want to ask the office if he prescribes natural thyroid hormone (Armour or equivalent). Most endocrinologists are more versed in diabetes than thyroid issues and often are TSH worshipers. I had to go to a naturopath to get the correct tests and NTH. There are a few places on the 'net to search for good thyroid docs--they seem to be few and far between.
BTW--the new range for TSH is .3 to 3.0, so you were definitely hypo when you got diagnosed!