Baby poos when rocking to sleep

Kate
It seems that almost every time my little girl is ready to be rocked to sleep, she poos in her diaper. Unfortunately changing her diaper startles and upsets her so much she becomes almost inconsolable until I feed her a bit again to calm her. But then the feeding causes another cycle of needing another diaper change. This is leading to some very long nights and I'm worried she's not getting the proper amount of sleep she needs. A couple of times I've just let the diaper be so she can get some sleep, but I don't want to irritate her delicate little skin. Is anyone else dealing with this? Or does anyone have any advice for me?