Let $G=\mathbf R$ act on the 2-torus $Z=\mathrm U(1)\times\mathrm U(1)$ by $g(z_1,z_2)=(e^{ig}z_1, e^{i\pi g}z_2)$. Lift the action to $T^*Z$ and use any $G$-invariant $H$.

Explicitly $T^*Z=\mathbf R^2\times Z\ni(p_1,p_2,z_1,z_2)$ where $G$ acts by the flow of $K=-p_1-\pi p_2$ (not proper by Bourbaki, Topologie générale, Chap. III, §4, Exercise 5), and say $H=$ any function of $(p_1,p_2)$.

(Comment $\to$ answer as requested.)

Let $G=\mathbf R$ act on the 2-torus $Z=\mathrm U(1)\times\mathrm U(1)$ by $g(z_1,z_2)=(e^{ig}z_1, e^{i\pi g}z_2)$. Lift the action to $T^*Z$ and use any $G$-invariant $H$.

Explicitly $T^*Z=\mathbf R^2\times Z\ni(p_1,p_2,z_1,z_2)$ where $G$ acts by the flow of $K=p_1+\pi p_2$ (not proper by Bourbaki, Topologie générale, Chap. III, §4, Exercice 5), and say $H=$ any function of $(p_1,p_2)$.