Let's think the hospital scenario through given what we've seen in We're Alive.

During Phase One of a zombie apocalypse you can expect the hospitals to get over run quickly and early. No matter how, or where the outbreak occurs, some of the infected are going to end up being transported to the hospital, where they are going to turn and spread the infection. The first wave of zombies, whether they come in through the doors or they come out of the emergency room, will over run the hospital and in short order it is going to be picked clean of human prey. The key to survival during this phase is to find a good place to hide and secure yourself. I'd suggest locking yourself in the most secure room you can find (radiology labs, store rooms, etc.) and douse the approaches with alcohol to hide your scent.

In Phase Two you can expect most of the zombies to leave in search of places where there is still prey. We saw this We're Alive at The Tower, and The Other Tower, where all but a handful of zombie hangers-on were gone by the time survivors searched the locations. Even Cain Memorial Hospital seemed to be occupied by just a handful of lurkers and malingerers up to the point that the zombie army arrived, presumably from the Stadium. This would be the time to search the hospital to gather supplies, look for other survivors and make your plans. Just be sure to arm yourself to deal with the lurkers!
For what it's worth, you might check home health care agencies during Phase One. Home healthcare workers (RNs, CNAs, etc.) get most of their supplies through the agency, which bills the patient's insurance or the patient him/herself directly. From there, you could take everything from gauze and hydrogen peroxide solution to morphine and saline bags (depending on what the agency has at the time of the attack.) They would have less than a hospital, but there should be enough dressings and such to last you through the initial wave. And unlike a hospital, most of the workers will probably abandon the offices asap. They usually aren't a great place to hide though. Many of them are just offices in strip malls or larger buildings.
Another good place to check would be a skilled care facility (as opposed to a retirement home, which probably won't have a lot of medications or dressings.) Most of the people in that facility are going to be physically impaired. So you could probably defeat them easily even if they are zombies. They will have a lot of pain medications, dressings for wounds, antibiotics and antibiotic creams, as well as blankets, sheets, and other stuff you'll need for daily life. Skilled care facilities for patients with Alzheimer's aren't a bad place to hide out, after you've secured them. They often have fences around them, not many windows that face out, and a central courtyard of some kind. They wouldn't be perfect but they could do in a pinch.

I don't know where else to propose this, so I'll do it here: I would try taking Rifampin or Sulfasalazine because both can change the color of your sweat. This might alter the smell enough to put the zombies off their meal as well.