 Solution

Properness is essential. The projectionis a submersion but is not proper. Its fiber over $x\ne 0$ is diffeomorphic to $\mathbb{R}$, whereas its fiber over zero is $\mathbb{R}\setminus \{0\}$ and has two connected components.

Merely requiring every fiber to be a submanifold is also insufficient. The surjective maphas every fiber finite and therefore a zero-dimensional embedded submanifold. Some regular values have three preimages and others have one, so the fibers are not all diffeomorphic. The map fails to be a submersion at $x=\pm 1/\sqrt{3}$.