I will say up front that my situation is a little different from yours, I have an autoimmune disease that has distroyed both my pancreas and thyroid. With the Dawn Phenom it is probably not about the medications you are taking. It actually happens to every person, it's just that if you are not a diabetic you don't see a difference. It is your liver releasing glucose to be changed into energy to wake you up. I don't know a single diabetic who has not dealt with this. I used to really REALLY struggle with it (so much that I had to go on an insulin pump). If it is decided that you need insulin to help with it, don't think of it as something horrible. Since it is just gestational diabetes it wouldn't be something that would go far past pregnancy, if you needed it at all after the birth. As for a connection, it usually goes the other way, as far as I can tell. Most people have problems with diabetes then their thyroid, but that being said, yes there are connections between the two. As for medications causing insulin resistance and higher blood sugars, that is also a major factor in the game. Depending on your medication needs they may not want to decrease anything that you need, but you may want to talk to your doctor/midwife about trying a decrease on the LevoT and see if they feel it would be a good idea or a bad one.
The only thing I can really suggest is don't fear the insulin. I know it's a needle, and it can be a little freaky giving yourself shots at first, but it wouldn't be the end of the world. I promise. With my first child I had to take insulin shots 5-6 times a day. (but again I was already a type 1 diabetic so it wasn't that much of a change)