I think that you could appreciate "Methods of Differential Geometry in Analytical Mechanics" by P.Rodriguez and M.deLeon (this is a link)

Apart from its intrinsec interest as reference for both the constructions of differential geometry and the geometrization of Lagrangian/Hamiltonian mechanics, in particular, if you are looking for the role played by the Poincaré-Cartan forms in mechanics, you could find it in their Chapters 2 (for the canonical almost-tangent structure on $TQ$) and 9 (for the Lagrangian Mechanics).

There you would find that to any (Lagrangian) function $L$ on $TQ,$ there are associated the Poincaré-Cartan forms $\theta_L:=J^\ast dL$ and $\omega_L:=d\theta_L,$ where $J:T(TQ)\to T(TQ)$ is the vertical endomorphism associated to the canonical almost tangent structure on $TQ.$

When the Lagrangian is not degenerate then $\omega_L$ is non degenerate, and the Euler-Lagrange vector field $\xi_L$ is the hamiltonian vector field w.r.t. $\omega_L$ with Hamilton function $E_L:=\mathcal{L}_{\Delta}L-L.$ (Here $\Delta\in\mathfrak{X}(TQ)$ is the Liouville vector field, i.e. the infinitesimal generator of the action of $\mathbb{R}$ throgh fiber-wise homotopies.)

**About the edited question:** Truly my historical knowledge is very limited, but probably the origin of these concepts could be in the works by Poincarè on the celestial mechanics, (when the concept itself of differential forms was germinating,) and therefore at the origins of the differential topology as we know it nowadays. You could look at the EoM entry on integral invariants and the references therein. (This is a link.)