= Solution
\b[Yes.] Choose a global <differential one-form> $\beta$ with $d\beta=\sigma$, and use the <cotangent fiber translation>
$$
T_\beta(q,p)=(q,p+\beta_q).
$$
It is a <diffeomorphism> with inverse $T_{-\beta}$, and $\pi\circ T_\beta=\pi$. Directly from the <Liouville one-form>,
$$
T_\beta^*\alpha=\alpha+\pi^*\beta.
$$
The sign convention $\omega=-d\alpha$ therefore gives
$$
\boxed{T_\beta^*\omega_\sigma
=\omega-\pi^*(d\beta)+\pi^*\sigma=\omega.}
$$
Thus $T_\beta:(T^*X,\omega)\to(T^*X,\omega_\sigma)$ is the required <symplectomorphism>. Equivalently, $T_{-\beta}^*\omega=\omega_\sigma$ gives the inverse direction. This <translation equivalence of exact twisted cotangent bundles> is global even when $X$ is not compact; no completeness argument for a <Moser theorem> flow is needed.
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