Left eye is bulging because of Graves and the eye is drying out a lot. What is the best treatment for this?

The reason for bulging of the eyes in Grave's is deposit of substances called glycoaminoglycans in the orbit, effectively making the eye socket too small to house the eye contents. If the bulging of your eyes is affecting your ability to achieve full eyelid closure leading the dryness of your eyes, first priority is to protect your cornea by applying lacrilube and artificial tears to your eye and taping your eyelids closed while you sleep. 

Second priority is to see an endocrinologist to get your hyperthyroid state in check. Sometimes this can be done medically, and other times it needs a surgical approach to remove the overactive thyroid gland.

Once this is done, a definitive approach to "decompress" the orbit can be done. In our practice, we perform both endoscopic medial decompression and open lateral decompression just depending on patient's particular case and anatomy. I hope you find this answer helpful.