I am a May graduate nurse from the accelerated BSN, RN program at Barnes-Jewish. Just like you, my ultimate dream and goal was to work in the NICU or with babies. It's fascinating to see those cute peanuts grow. Also, I felt the technology was so incredible; I'd be learning throughout my career. After 5 months of job searching, I was fortunate to land a position in the NICU at Cardinal Glennon. I'm in my second week of orientation and loving it!!
I was in a year program, so I didn't have time to be a tech. I suggest to shadow, shadow, & more shadowing. I did at least 4 times. I am not sure if Children's, St. John's, or St. Mary's have techs in their NICUs. If they do, I would definitely get in there! However, at Cardinal Glennon, we only have RNs representing the nursing staff on the floor. So, shadowing in their unit or others would be your best bet. Not only does shadowing show your initiative and interest, but you can become familiar with the machines. Also, you can get some contact info from managers or other RNs.
Best of luck to you and please let me know if you want more info!!