Persistence Ignorance

Your questions and doubts ring an interesting alarm here, I think you went a bit too far in your interpretation of a "rich domain model". Richness doesn't go as far as implying that persistence logic must be handled by the domain objects, in other words, no, they shouldn't know how to save and delete themselves (at least not explicitely, though Hibernate actually adds some persistence logic transparently). This is often referred to as persistence ignorance.
I suggest that you keep the existing DAO injection system (a nice thing to have for unit testing) and leave the persistence layer as is while trying to move some business logic to your entities where it's fit. A good starting point to do that is to identify Aggregates and establish your Aggregate Roots. They'll often contain more business logic than the other entities.
However, this is not to say domain objects should contain all logic (especially not logic needed by many other objects across the application, which often belongs in Services).

 ian31 - http://stackoverflow.com/questions/4708689/ddd-where-to-put-persistence-logic-and-when-to-use-orm-mapping