This whole issue stemmed from your arrival late to the gate--which was caused by a Travelocity ticketing issue.
The GA was rude and you should write a complaint about that.
But the fact that traveling internationally with 3 children---you put ALL your personal items in a bag that doesn't fit under a seat---that is not good planning.
I never depend upon finding over head space---that is NOT a guarantee and flights are often full.
Whatever is needed for any of the family members (clothing, diapers, baby food) should be in a carry on that fits under the seat. Each child has their own small backpack with their change of clothes, snacks, entertainment. Adults have a soft bag with wallet, passports, phone, medications, etc.
Plus---you can't access the roller board during take off and landing, or if there is turbulence.
Additionally, checking in online 24 hours prior would have probably given you a heads up that there was a ticket issue. Which may have been resolved over the phone (either with AA or Travelocity) prior to airport arrival. If not---then you know to get to the airport at least 3 hours prior (which I would recommend for international anyway) because you have a ticket issue.
When you calm down---look at the root cause and determine what you can do next time to avoid issues.