Yep, it's a total replacement of the drywall... really it's the easiest and best way to go.

Can you take a picture of what you are talking about and post it? I'm guessing it may be something other than caulk (maybe grout) but without a picture, it's hard to tell. You can post a picture by using imgur.com and adding a link to your post/comment.

Generally speaking, when I have a paint question, I'm not sure of, I go to a local Sherwin-Williams store and tell them what materials (the MDF in your case) I'm using and what sheen finish I want.

If you don't want to do that, then as others have said, an oil/solvent base primer is what you want and both Zinseer and Kilz make it specifically for MDF. If you have to deal with edges, it's a good idea to use spackling on the edges, sand smooth then prime. Edges of MDF are porous and soak up paint and don't always give you a good finish.