I'll take a year to apply. Should I search for new logic projects or round up what I have?
Given that a doctorate is about depth in a small area, your option 2 sounds better to me as long as you have sufficient breadth of knowledge in your field already. 

Note that many (most?) US students move to a doctoral program directly from their undergraduate program, so the "breadth" required is only about that required in a baccalaureate. So, I think, that except for the time factor you mention, you are likely well prepared now. But getting a bit of research going certainly won't hurt you, even if you have only minor results. You probably won't be working with a guide, after all. The area in which you do the research, however, might matter. It is better if it is closer to the sub-field that you actually want to work in.