I feel for you. I also really hate needles; luckily I don't pass out but I can't stand the sight of them either. Annual flu shots are dreadful, but I still get them. When I had to give my cham his first injection, I picked up the needle and my heart rate sky-rocketed, my breathing became shallow, but I could still think properly. I calmed myself down, walked over to him and gave him the injection. After giving him several shots over this past month, my breathing and heart rate don't change any more. While I still dislike needles, I can tolerate them much more than before my cham's incident.
I wouldn't worry about taking the dripper out of the cage as long as she goes for it on her own volition (whether that be from a leaf or straight from the dripper). Somewhere along the way, I missed the cues that he was done drinking and wanted a breath of air. As long as the dripper isn't spraying water like a firehose, and she has the ability to walk away from the water whenever she wants, the dripper should be perfectly safe. The dripper itself wasn't the cause, I was. We live and learn, and sometimes we make mistakes that hurt the very animals we vowed to protect. Unfortunately, that's the way of life. As long as you can provide them the best care possible and learn from your mistakes--well, that's all that our pets could ever want.