A bit of clarification: you may have not noticed that I mentioned that my double vision corrected during the cold phase, to the point where I only have it when I am very tired or “when I am hypothyroid.” Having wonky thyroid levels does interfere with the functioning of our bodies, including those eye muscles responsible for TED. The distinction is not so much whether synthroid is a causative agent, but whether or not we are on an appropriate dose of it to keep our thyroid levels normal.
If you are having trouble getting a dose of replacement hormone that provides long-term stability, then perhaps some trouble-shooting would help. Keep in mind that when we are first going onto replacement, there can be some months of tinkering with the dose. Making minor adjustments should not have an impact on your eye situation, but if you are frequently going hyper or hypo, then perhaps some of the following ideas might help:
Make sure to take your synthroid first thing in the morning, on an empty stomach, with a glass of water, and wait a minimum of half and hour or forty minutes before eating or taking other medications.
Do not take vitamin or mineral supplements until four hours after you take the synthroid.
If you are taking other medications, talk with your pharmacist about how best to space them with the synthroid.
The issue is to be as consistent as humanly possible with the taking of the replacement hormone so that a consistent dose of it is absorbed into the blood. Once it is absorbed, it does not matter what you eat or drink, but getting it into the blood in a consistent manner can make a difference.