I just tested this locally and it works as expected, so there must be something specific to your install.
I'll probably need a copy of your site (hidden link) but before preparing that I suggest you check your browser console for errors, and also your PHP logs.
If you haven't already, turn on the debug log by editing your wp-config.php file and change the line with WP_DEBUG like so:
That will create a file debug.log in your /wp-content/ directory which you can examine in any text editor. Try again on the post edit screen to re-order and save the RFGs and then inspect the log. If you don't find the debug.log file it means it didn't generate any warnings or errors.
Also, it looks like your version of Types is slightly out-of-date.
Instead of using the automatic update, it might be a good idea to delete the current Types, download a fresh current version from toolset.com/account/downloads, and then upload the zip file on the add new plugins page.
I'll mark your next reply as private in case you have a copy of the site to share.
The issue appears to be related to the conditions I've set on the fields in the RFG. If I remove all the conditions the problem is resolved.
Most of the conditions relate to a field outside of the RFG. We wanted to hide the fields in the RFG if they weren't applicable due to the values outside of the RFG.
A quick update to let you know a fix for this will be included in the next Types release (3.1), which is about to enter QA testing and so should be released if not next week then the following week, all being well.