In the case where $ F \subset T^*(M) $ is open the argument is roughly as follows: locally closed 1-forms on $M$ are in one to one correspondence with Lagrangian submanifolds of $ T^*(M)$ (with its canonical symplectic structure) which are transversal to the projection to $M$. Given a point $p\in F$ you can then choose a Lagrangian plane $\Pi \subset T_p(T^*M)$ transversal to $\pi$ and a extend it to a Lagrangian submanifold through $p$. This will be still transversal and contained in $F$ after restricting to a sufficiently small portion.
I'm a bit in a hurry at the moment, but you can ask me for more details or probably find them in any text on symplectic geometry.
