Forcing with copies of the Rado and Henson graphs

If $B$ is a relational structure, define $P\left(B\right)$ the partial order of all substructures of $B$ that are isomorphic to it. Improving a result of Kurilić and the second author, we prove that if $R$ is the random graph, then $P\left(R\right)$ is forcing equivalent to $S⁎\dot{R}$, where $S$ is Sacks forcing and $\dot{R}$ is an ω-distributive forcing that is not forcing equivalent to a σ-closed one. We also prove that ${P(H}_3)$ is forcing equivalent to a σ-closed forcing, where $H_3$ is the generic triangle-free graph.