Shane, I don't think you have a problem, there is a lot on your plate right now and there is no doubt stress makes a person feel more vulnerable than usual. There are times when sensitivity to a situation will make me decide not to carry and during those times I know that my options for self defense will be less. You may opt to have the gun packed in the suitcase in case all hairy carry should break out, it could still be an option. For the lesser threats you may be able to outwit and outplay.
I don't think the sensationalism of Osama being the attending or kidnapping would top my list of real threats that are possible in a hospital situation. That is just an inflammatory insult to your intelligence. I would be more concerned about a psychotic patient or maybe a disgruntled worker returning to take revenge. Sometimes the emergency departments are located close or within proximity of the regular patient wards. For the sake of security and so you can concentrate on Mom and baby, know ahead of time where you will be, where the exits are, waiting areas, stairwells, and just the general layout of the facility.
I know of at least one member here that carries at Gundersen in LaCrosse with no problems to date. I considered it as well for pt but I really felt that various exercise equipment didn't offer good safe muzzle control.
Many Blessings at this great time of your lives.