Hi Alexia:
I have some answers for you.
Discrepancy between LOF numbers and source file numbers
The first one is addressing the mystery of why the figure numbers increment correctly in the LOF but not on the page. The reason is because you are not pulling in the para numbers from the source files—instead you are adding para numbers to the LOF para tags on the reference pages. This is ok to do, but it was causing confusion in the thread.

To pull in the auto-numbers from the source files, remove the auto-numbering on the LOF tags and add <$paranum> in front of <$paratext>:

Update your book. Now the source file issue is represented in the LOF:

Incrementing the numbers in the source file
The problem isn't in the Body.fm file, or with your Continue Numbering choices—you did all of that correctly. The issue is actually in the Cover file and you can see it in the screen shots below. The reset is happening on page iii, before we ever get to Body.fm.

There isn't another paragraph tag forcing the reset, instead it is because the frames are disconnected, and in FrameMaker, the counter starts with zero for each new flow. The fix is to navigate to page iii, select the text frame and choose Format > Customize Layout > Connect Text Frames to connect it with the frames on the previous page.

Update your book and take a look at the LOF. Everything is now incrementing correctly in both the LOF and in the source files.

NOTE: Once you have this working and have some free time, I would suggest creating a new custom master for page iii, based on page I (auto-connect will be on as a default) and then apply it to page iii. Page iii is using a sidehead column, which I don't see the need for, so you can clean that up at the same time. Back up the original cover file before you start so that you can restore it if you run into trouble.
As a general rule, we don't want use disconnected pages unless there is a specific need to use them.
~Barb