I can tell you that will never happen. The human body is composed of water and people want to resupply it everyday. At the same time, people undergoes a process of metabolism. Water takes useless substances from human's body through sweating,urinating,etc. Weeping is also one way of matabolism. Therefore, when the day you quit drinking water comes, you will run out of tears.
No, you won't.
As long as your tearing system continues to work and there is nothing wrong with it, you can produce tears endlessly. If you develop dry eyes sydrome, there may also be a lack of tears in you, too. But it seems that this is not your case at all.
Well, in my opinion, it is possible for your eyes to have tear running out. So you do not need to worry about it. On the other hand, for your watery eyes, it can be annoying and dangerous. First, you should know that watery eyes can be a sign of dry eyes. 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. At that moment, the tear glands react by producing large amounts of tears. And that can lead to some symptoms which just like you had. Also, in some cases, allergies and eye infection can also cause watery eyes too. So just check your situation as soon as possible.