The situation with infinite rank edge and vertex groups is qualitatively very different. For example, the following paper by Hagen and Wise shows that a non-trivial class of hyperbolic groups (constructible groups) are virtually HNN extensions of (infinite rank) free groups Special groups with an elementary hierarchy are virtually free-by-ℤ.
On a technical note, in Swarup's original paper, the paper by Diao and Feighn mentionned in another answer, and my own paper On the one-endedness of graphs of groups which is inspired by Diao-Feighn, finite generation of the edge groups plays a critical role. In the more modern papers, finite generation of edge groups ensures that the group acts co-compactly on a nice square complex.
All this to say that a generalization of Swarup's Theorem to your situation may require to be some new ideas.