I applied for a bunch of dissertation completion fellowships. I earned one national fellowship and am on a waitlist for another national fellowship. I wasn't required to submit a dissertation chapter for either application, but I did have to submit my dissertation prospectus, which was quite rough. I think my application was successful because I articulated how my research fit with the funding organization's mission. Maybe it goes without saying, but it's not enough, as I've learned, to articulate the obvious--that the dissertation fellowship will help you complete your dissertation and become a scholar. They know that already; you've got to convey the "how" of it.
If your university offers a grant writing workshop, take it! Your proposal has to be clear, concise, and attractive. I've been told to use a hook when writing the proposal; however, I don't think my proposal had a hook. Anyway I benefited from workshopping my proposal and other application materials with a writing group. They were just the right kind of audience--non-specialists, intelligent readers. They reminded me to go back to the basics when crafting the proposal.
Also check out this thread on the Chronicle of Higher Education website. More suggestions are provided there.