= Solution
For any positive-dimensional $X$, choose a smooth function $h$ with $dh$ not identically zero. A nonconstant smooth bump in a coordinate chart supplies one on an arbitrary <smooth manifold>. Translate by this <exact differential form>:
$$
T_{dh}(x,p)=(x,p+dh_x).
$$
The <cotangent fiber translation> identities give
$$
\boxed{T_{dh}^*\omega=\omega,\qquad
T_{dh}^*\alpha=\alpha+\pi^*(dh)\ne\alpha.}
$$
Thus a <nontrivial exact fiber translation> is a <symplectomorphism> that fails to preserve the <Liouville one-form>. For example, on $T^*\mathbb R$ one can take $T(q,p)=(q,p+1)$: it preserves $dq\wedge dp$ but changes $p\,dq$ to $(p+1)\,dq$.
If dimension zero is admitted, $T^*X=X$ and $\alpha=0$, so every <diffeomorphism> preserves $\alpha$ and this requested distinction is impossible. The example therefore applies in positive dimension, as required for a nonzero <differential one-form>.
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