It would help if the pictures weren't so big. Hard to take in the problem. Don't throw out your old shoes, they can be re-lined.
Hard to help without some sort of photos or information,,but I saw this on another members photo site. You might check in with JeffB2's 52-59 Social Group here on the hamb or Join the '54 Ford Club of America,, www.1954Ford.com. Hopefully with all the combined knowledge with both groups someone has experienced your problem and can offer a solution.HRP
You are correct about non adjusting brakes,,but not having a clue as to what the problem is I offered what I consider a solution,,the thread is here: HRP http://www.jalopyjournal.com/forum/group.php?do=discuss&group=&discussionid=20141
Here's the only picture I have of my rear brake set-up... I was actually taking a pic of the backing plate but maybe it will help.