Conformal-symplectic geometry ?

If the manifold has dimension bigger than 2, I think the conformal class of $\omega$ is just $k\omega$ for constants $k$. Locally, by Darboux we can write $\omega = \sum_i dq^i \wedge dp_i$. If the dimension is greater than 2, the only way for $0 = d(f\omega) = df \wedge \omega$ is if $df = 0$.

There is a notion of conformal symplectic structure related to what you are asking. I refer to locally conformally symplectic manifolds. These are manifolds $M$ equipped with a non-degenerate two-form $\omega$ and a good open cover $\left\{ U_{a}\right\}_{a\in I}$ such that for every $U_{a}$ there exists a function $e^{f_{a}}\in C^{\infty}(U_{a})$ satisfying

$d\left( e^{f_{a}}\omega|_{U_{a}}\right)=0$

This is equivalent to the existence of a flat real line bundle $L\to M$ with connection $\nabla$ that descends to a well-defined closed one-form $\varphi$ in $M$ satisfying

$d\omega + \varphi\wedge\omega =0$

One can define the coboundary operator $d_{\varphi} = d +\varphi$ on the complex of forms $\Omega^{\bullet}(M)$, whose cohomology is the so-called Lichnerowicz cohomology, which in general is not equivalent to the standard de Rahm cohomology. The two-form $\omega$ satisfies $d_{\varphi}\omega = 0$ and it is thus a cocycle. For further information you can check Izu Vaisman's papers from the 70's and 80's on locally conformally symplectic and K\"ahler manifolds.

Yes, this has been considered (hasn't everything). See the following antique reference:

http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1978__19_3/CTGDC_1978__19_3_223_0/CTGDC_1978__19_3_223_0.pdf