Property FA is equivalent to your property for groups $H$ that do not decompose as a free product. Indeed, suppose that your property holds but $H$ acts non-trivially on a simplicial tree. Then $H$ decomposes as a non-trivial amalgamated product $A*_CB$ (it is either that or an HNN extension which your property rules out). That is $H$ is not conjugate to a subgroup of either $A$ or $B$. Consider the free product $F=A*B$ and an HNN extension $E$ of $F$ conjugating two copies of $C$ there (one in $A$ and one in $B$) with free letter $t$. Then $A*_CB$ is isomorphic to the subgroup of $E$ generated by $tAt^{-1}$ and $B$ (it is proved in Lyndon and Schupp). Hence $H$ is a subgroup of $E$. By your assumption, $H$ is a subgroup of $F$. Hence by Kurosh's theorem $H$ is a non-trivial free product. 

For groups which are non-trivial free products, these properties may not be equivalent. For example, consider $PSL(2,\mathbb{Z})$ which is a free product of two cyclic subgroups. I do not see now how to embed it into an HNN extension without being a subgroup of the base group.