My daughter had clogged tear ducts in on eye since birth, so we had to deal with goopy green gunk a few times/day for about 10 months! It really is just the duct between the nose and the eye. I didn't know this, but when you cry, and your nose gets all runny, it's because your tears drain down your nose. When kids get stuffy, their ducts get blocked and backed up. They get green snot out their nose, and whatever's backed up goes out their eye the other direction. Gross - but it is just snot - not pink eye - as your doctor told you.
The best solutions we found were the warm damp cloth to loosen the goop every few hours. Also, our doctor showed us how to massage the duct. You can probably find it described online, but you can feel a tiny "ball-shaped" duct between the corner of your eye & the bridge of your nose. Gently press that spot a few times, and sometimes, you can get the duct to push some of the gunk out & clear itself. Otherwise, treat the cold symptoms - you can use the "little noses" nasal saline spray and nose syringe, a humidifier, etc. If she continues to have it after the cold goes away, go back to your doctor. There are a large number of kids who are born with particularly small ducts (like my daughter) that are chronically blocked. Most will grow out of it, but some will require a quick, simple surgery to unblock it. :-(
Congrats on your baby turning 1! :-)