I’ve finally figured out the problem. All the code, in all it’s variations was fine and works, except for one tiny thing. The variable ‘$name’ is used by the WordPress system to display the blog name, and as I was using that as my variable for the name field on my form it was causing issues. Who knew.
Anyway, I simply changed it to use ‘$fullName’ and changed the id and name attributes on the form to ‘full-name’ and everything worked great.