I guess you forgot to deactivate Live Link for the MH body after recording , it is working fine here (UE5.2.1) , see video ( showing FBX import and Live Link recording ) :
Also be sure you are using the latest versions of CC_Setup , IC Live Link and Live Link MH Kit from Reallusion.

Thanks Style Marshal! I'm a big fan! I have tried everything.  Here are links to the project files and an FBX https://drive.google.com/drive/folders/16Jko3vxPmwgUuroKtqUU4JDBWhrauVK4?usp=sharing
And here is the Unreal Project  https://www.dropbox.com/sh/74zaq1fjdp53oty/AAAVW5W1WB3sDBtkh670cB6_a?dl=0

I seriously think something is goofy because I used Motion Director to animate the walking. Then added the arm animation. Then flattened it. On the project named "Mocap_Fix" I zeroed out the animation to start at the World Origin point, but it just keeps messing up.

 I'm going to try using the motion file on a different body type. I'm also going to try starting a new project file and applying to the same body type again. I may have to uninstall and reinstall iClone 8. I already uninstalled and reinstalled UE5.2.1 yesterday. 

I did the same process on this video. And the shoulders and arms keep getting wonky.

  

Don't know what went wrong in your project , could be the MH Version ( older or upgraded ) but it is easy to fix :

OMG! Thank you so much. I seriously need to learn more about the retargeting system. I can't believe it was that simple. Thanks!!

Is there a way to export FBX facial animations to Unreal Engine for Metahumans?

Yes but I would recommend you the Live Link recording , it is easier and looks better.
If you don't have Live Link or CC4 only you can use this way but you will need the MH Live Link Kit , it is free ... Here is what to do for CC facial import to MH via FBX :