How do I rearrange app icons in the dock?

Boot into Safe Mode (How to use safe mode on your Mac - Apple Support) by booting with the Shift key held down and tr there to see if the problem persists.  Reboot normally and test again.

NOTE: Safe Mode boot can take up to 10 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

Just click and drag. Folders go on the Trash bin side t of the dividing line and apps on the other side of the line: