Can anyone recommend a DAM with a centralized database

Extensis is the best for what you want to do. It is not a set it and forget as you will need to train and monitor everyone to insure that they actually make the change and don't have multiple sets of files on the database and on their personal computers as well. It is not cheap but you can use the price as a hammer to help force compliance.

I would start with one department and work out the kinks before expanding it throuughout the hospital and get a key manager as the project sponsor and champion.