This is happen to most of people. You don't worry about it. It is just normal reaction of your body. As we know, our eyes tearing is just to describe our emotion. When we feel happy, sad or moved, we will get our eyes watering and can't control it. So, It is normal for you tearing when you feel embarrassed. What you should to do is try to ignore such feelings that may help you.
Usually the embarrassing situation will not cause the eyes start water. Your situation is special. From my basic judge, I think your eyes get the infection from the dust or other things. I suggest you to go to see the doctor, have a full check on the eyes and accept the treatment.
Well, from what you describe, I can see that your eyes are allergic to special occasions, sometimes people will have some allergic reactions, for example, I know someone who will sneeze when encountered some special things. Perhaps you are just a little bit shy and are not used to get embarrassed.You should know different people have different situations.
Tear contains enzymes which could protect your eyes and nasal mucosa from bacterial infection. It is a self-defense method of your body. When you feel some emotion impact, the tear gland will receive the order from brain, the tears will be produced. When people feel agitate or other feeling, the capillaries will congestion by blood, the muscles around eyes will contrast, the tear is the by-products of this movement. It is normal and let it be; the tear will take away most of the pressure on your emotion, and keep your emotional health. However, if you always feel anxious and embarrass, it may be the no relation with the eyes but your emotion, you can go to psychologist for check.