When I was pregnant with my first I had Bell's Palsy. It's a viral infection that effects the facial muscles. I could feel the left side of my face, but I couldn't move anything. It was very weird.
I didn't have any trouble keeping my eyes moist, as my eye just kept watering. It is very common for people to not be able to shut their eye all the way, especially when sleep.
The doctor in the ER as well as my own OB and every doctor I have talked to about it since then mentioned taping my eye shut.
Do not be afraid to do it. I did it the first night. My husband did it for me with some scotch tape (I would not use duct tape). The next morning he ever so gently removed it. If it were to hurt at all, I probably would have just taken it off in the shower.
It's really worth a try. Dry eyes are not something I would want to mess with. I thought the sleep mask was a great idea. You could definitely try it. Just keep in mind that taping your eyes shut at night comes highly recommended by many doctors.