If the neck screws won't simply push through the holes in the body, I'd strongly advise drilling them out so that they are a tight push-fit. You may not get rid of the gap fully, but the neck will be as firmly attached as it can be.

And FW's suggestion of a string retainer bar (just a few $) is definitely worth it. A locking nut like that won't work properly without the strings being pulled right down onto the headstock edge.