What is the best way to resolve issues with AWS services that are not working properly?

I am asking this question in general, though I am motivated by my following (hopefully representative) problem.

I've been trying to obtain a domain authorization code from AWS in order to transfer to a new domain registrar. However, requesting such a code just returns the error

   "Sorry, but an unexpected error has occurred while getting your auth code. Please retry again..."

This error has lasted for over a month. I've posted on the AWS forum [1], and there was no resolution though some other commenters expressed having the same issue.

Of course, since I am not paying for an AWS service plan, I don't expect dedicated customer support. However, the service provided is not meeting the basic requirements. In fact, since I cannot obtain the authorization code, I cannot even transfer the domain to another registrar (which is the other response to an unsatisfactory product!)

[1] https://forums.aws.amazon.com/thread.jspa?threadID=347652