How can I move from chemical engineering into drug discovery?
As I mentioned above, drug design is not really a "major" field, in the sense that it's rare to be able to get a diploma which lists "drug design" as the field of study. It's a dissertation topic or a focus area. 

The relative advantage of this is that very few people come in with experience in the field, because it's so specialized. That means it's harder to be "picky" in selecting students—if everybody has to have lots of experience, you may not end up with enough qualified candidates.

Given your past experience with drug delivery, you probably have a fairly good idea of some of what is involved in drug design, so it's not as if you're coming from a completely unrelated discipline.

So, I don't think you have much to lose by applying to PhD positions while you look for alternate opportunities to work in the field. (Sometimes you need to apply more than once before you get accepted!)