I’ve managed to edit on of my part layouts to my liking, but unfortunately these were not global edits and affected just the one part. I don’t want to have to go into each individual part and add text boxes, make resizing edits, etc for each individual part. How may I propagate the layout changes from this one good layout to all the others?
You need to make the relevant changes to the page templates (or master pages in 3.5) themselves, not the individual pages of a part layout. You can’t propagate the changes from the individual pages of a part layout back to the page templates (master pages).
One thing you could do, I suppose (edit: but only in Dorico 4, not 3.5), is delete all the other existing part layouts, duplicate the one part layout you’ve already tweaked (from the context menu in the right panel of Setup mode), then manually unassign the existing player and reassign each of the other players to the duplicate part layouts. If the work you’ve done to the one part layout is extensive this may be the quicker option.