There is no need to make them overlap. They just need to be adjoining and you can't make them adjoin properly unless the parent is a block box (which floats will create nicely). You can still create separation by adding padding-top to the submenu which creates the illusion of space while in fact still keeping contact.
And when I move the submenu items up, they only work when they all overlap the parent, which isn't the most aesthetically pleasing thing in the world :P
But I'll work on it. Thanks so much for your help!