This also happened to Paula Jones.
You are precisely correct. MOC Karen Thumond (D-FL) had the whole story from the Martins when she accepted the tape from them. She KNOWINGLY transported illegally-obtained material across state lines. She gave it to McDermott, who then KNOWINGLY provided illegally-obtained material to members of the press in a blatant effort to skew the public's impression of the contents.
It is an outrage that all these MOC's are using the Speech and Debate clause in this manner. The Founding Dads put it in specifically to keep politically-motivated arrest, seizure, etc., from being an obstacle to good government, NOT to allow these jerks to violate the law willy-nilly and get away with everything possible.
Wiretapped Conversations by Kathleen Kirby, for Communicator, February 2000
This article directly references the Martin case as well as others.