"all torn up": I was making a pun on tear as in teardrop and tear as in rip up.
When eyes well up they fill with tears.
For information on people who have very copious tears unrelated to emotion you should probably consult physicians rather than us. There is no ordinary English word for that phenomenon.
I think the term you are looking for is watery eyes. Sometimes you hear runny eyes but it is more related to allergies and fevers.
It's not commonly used unless you are describing the condition, and yes no emotion is involved.
Do an online search of the term and you'll get more examples of how it's used.
Not a teacher!
@SanMar: I think the term you are looking for is watery eyes.
Yes, as in: "My eyes start watering whenever I yawn", or "Just thinking about chopping that onion makes my eyes water".