Timo2, UpAnchor - Thanks for the suggestions and answers.
I'm all set up with 2FA on MySocialSecurity. What I did was ending up paying a visit to a Social Security Office. The person pretty much just looked and my driver's license, asked me to verify my email, then printed out a piece of paper with instructions for signing up.
I'm feel so much more comfortable now with the extra safety.
The rejection for online verification my very well be from me adding a fraud alert on my credit report. Now I've put a freeze on my credit too. Shields up!