This problem was making me nuts. I so like moon reader but the sync feature has to work. What I discovered is in the latest release 1.8.8 it appears the way the dropbox sync has changed. It has a new folder structure on dropbox.
What I did to fix the sync was to make sure all my device were running the same version 1.8.8. My phone had not been updated which was the first of my problems. My tablet had been updated but sync and backups were not making it to dropbox. So under local files I selected dropbox at the top to be sure it was linked. It was but backups and sync was still not working. While still in local files I selected the settings gear symbol and then selected unlink dropbox account. After that, I selected dropbox at the top of local files again and moonreader went through the linking process.
This is how I fixed my broken sync. I did multiple tests with multiple books. I always need to remember if it ain't broke don't fix it by updating apps. Updating software and apps has caused me more grief than the new features were worth.
Hope this helps others and they do not give up on moonreader, it is the best reader out there.