The principle is simple, when your eyes water, the water is covering in the surface of eyeball, and it become the transmitter between your eyes and air. The water's refractive index is totally different from the air. When the rays reflected from the object of aim which you are watching, it contacts with the tear water at first, and then gets into your eyes. It changes the process of imagine from your eyes. So the double vision is formed. It is normal, but it may hurt the eyes and vision if the watery eyes exist for a long time. Go to hospital to heal your eyes at first please.
Well, in my opinion, it is possible to have double vision when you suffer watery eyes. And for watery eyes, it can be a sign of dryness. Generally speaking, tears are for lubrication and for keeping your eyes clean. And when your tear glands don't work in a wrong order, it may develop dry eye syndrome, leading to watery eyes. At that moment, some other symptoms may occur, such as double vision. So you need to go and see the eye doctor as soon as possible.
Yes, it is normal for you to have double vision because of watery eyes. The watery eyes may mean that your eyes get infection from the invisible bacterium. You will see things not clearly, in the blurry state. You'd better use the warm compress to make your eyes feel comfortable and moisture. You could also use suitable amount of eye drops to make the eyes feel comfortable and see clearly.