Going to need some specifics on this one.  I was not able to replicate the problem based on this info.

I tested the Marriage page both as a full admin, and an "Office Page Author", which does not have Publish Pages capability.  In both cases, the ACF elements including calendar, buttons, faq, contacts, and mondrain tiles displayed as expected.

If there is an issue, it doesn't seem to be tied to Publish ability.  Let me know which user(s) are having the problem, and I can hopefully find a common denominator.

If you are adding new items to the page, then they do not show up in preview. If they have already been published, then they do display in preview. 

For example, this page was previously blank (no title or content at all) and I added some different areas. If I want to see what it looks like before submitting for review (can't publish) then this the second image is what I see in the preview. 
https://www.dioceseofscranton.org/50087-2/

Once I marked it as "Pending Review," it did update the preview page.

Is there any way around having to save it before previewing it? 

Veronica - Circling back to this one, and again I've managed to not be able to replicate the symptoms.  How were you able to replicate them?  What user account and/or role were you testing with?  I tested with Office Page Author, which does not have Publish ability, and tracing your steps the Preview still showed the correct.  

Seems to be working fine now. Previously, I could get everything but the mondrian tiles and the calendar to work. 

Not sure what I was doing before, but I'm glad it's working now.