I want to update on our situation.
We waited a full three weeks after the EAD.
Went to the SSA office (bring your EAD card, passport, and patience since the wait could be 2-3 hours at the office).
Some office need appointments, our did not. Check their web site.
The SSA said the info didn't get transmitted to them so they put in a new application. SSN card arrived in the mail less than one week after that.
We saw people getting into arguments with the security guards at the SSA office, don't do that. Just have some patience, they are short of staffs.