Her sons Nicholas and Nathaniel are 2 weeks old now, and doctors say their chances of survival are good, but only if they receive heart transplants within the next three to six months.
While the boys remained hospitalized at UCLA's Mattel Children's Hospital, Draper and her husband, Michael, pleaded Monday for organ donations that could save their lives.
"We want our boys to have a chance," said Michael Draper. "We want them to come home."
Home is Phoenix, where Michael Draper works in admissions for the University of Phoenix.
Soon after they were born on July 11, Nicholas and Nathaniel were flown separately to UCLA. Their parents have only been able to hold each of the boys once since their births.
"It's just hard," said Michael Draper, 33. "When you walk into that setting it's hard to say, 'Let's just spend some time with our boys and pretend everything's normal."'
The Drapers, who are both registered organ donors, said they don't want other families to suffer the death of a child. But in the event that a baby does die, they hope the parents would be willing to donate the organs to help them or other parents.
Nicholas is already on a list of prospective recipients. His brother, Nathaniel, will be added soon. They will receive hearts in the order that they become available.