Suppose that we have a smooth variety $X$ of dimension $n$ that fibers (a flat morphism) over a curve $Y$, and s.t. the fibers of $X \to Y$ are all complete interesections of two quadric hypersurfaces of dimension $n$.

Suppose that generically the quadrics are smooth and the singular ones have at most corank 1.

It is easy to see that such an object always has a rational section Y->X. Does it always have a *regular* section? Maybe for $n$ big enough? By regular I mean a section with no intersection with the singular locus of the fibers.