Remove trailing whitespace from a field
Okay, here is what finally worked: wrap every individual field in {% spaceless %} (not just the whole block) and use render and trim together.  Trim alone makes the variable disappear. Here's my markup:

&lt;p&gt;
  {% spaceless %}{{ content.field_story_program_name|render|trim }}{% endspaceless %},
  {% spaceless %}{{ content.field_story_relationship|render|trim }}{% endspaceless %},
  {% spaceless %}{{ content.field_story_city|render|trim }}{% endspaceless %}, 
  {{ content.field_story_state }}
&lt;/p&gt;


Extra thanks to @Cyclonecode for giving me a variety of tricks to try.