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I figured out the problem. In Layouts > Custom Margins > Layout
I had “Section start:” ticked for “Odd Page.” When I set it for “Continuous” everything started to work normally. I want Parts and Chapters to start on odd pages and they seem to be stable enough if I go into Layout > Breaks and set them individually, manually. For some reason, when it’s set globally for “Odd Pages” things become very unstable when you try to set individual sections for "continuous". This appears to me to be a bug that needs correction.
Thanks for your time and attention.
EDIT: I was wrong. The sections I want on "Odd Pages" will not be stable unless I highlight those sections, then go into "Layout > Custom Margins > Layout and set them for "Odd Pages." being sure Apply to "This Section" is selected. The individual settings available in Layout > Breaks will revert to "Continuous" otherwise. This isn't at all intuitive. Some redesign seems called for. Obviously, using the Layout > Customs Margins > Layout overrides the Layout > Breaks settings, which should be greyed out if the Layout > Custom Margins > Layout is in effect, with an explanation.