Let (M, g) be a closed connected orientable Riemannian manifold of dimension n ≥ 2. Let ω: = ω0 + π*σ denote a twisted symplectic form on T*M, where \({\sigma\in\Omega^{2}(M)}\) is a closed 2-form and ω0 is the canonical symplectic structure \({dp\wedge dq}\) on T*M. Suppose that σ is weakly exact and its pullback to the universal cover \({\widetilde{M}}\) admits a bounded primitive. Let \({H:T^{*}M\rightarrow\mathbb{R}}\) be a Hamiltonian of the form \({(q,p)\mapsto\frac{1}{2}\left|p\right|^{2}+U(q)}\) for \({U\in C^{\infty}(M,\mathbb{R})}\). Let Σk: = H−1(k), and suppose that k > c(g, σ, U), where c(g, σ, U) denotes the Mane critical value. In this paper we compute the Rabinowitz Floer homology of such hypersurfaces. Under the stronger condition that k > c0(g, σ, U), where c0(g, σ, U) denotes the strict Mane critical value, Abbondandolo and Schwarz (J Topol Anal 1:307–405, 2009) recently computed the Rabinowitz Floer homology of such hypersurfaces, by means of a short exact sequence of chain complexes involving the Rabinowitz Floer chain complex and the Morse (co)chain complex associated to the free time action functional. We extend their results to the weaker case k > c(g, σ, U), thus covering cases where σ is not exact. As a consequence, we deduce that the hypersurface Σk is never (stably) displaceable for any k > c(g, σ, U). This removes the hypothesis of negative curvature in Cieliebak et al. (Geom Topol 14:1765–1870, 2010, Theorem 1.3) and thus answers a conjecture of Cieliebak, Frauenfelder and Paternain raised in Cieliebak et al. (2010). Moreover, following Albers and Frauenfelder (2009; J Topol Anal 2:77–98, 2010) we prove that for k > c(g, σ, U), any \({\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}\) has a leaf-wise intersection point in Σk, and that if in addition \({\dim\, H_{*}(\Lambda M;\mathbb{Z}_{2})=\infty}\), dim M ≥ 2, and the metric g is chosen generically, then for a generic \({\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}\) there exist infinitely many such leaf-wise intersection points.