After creating a new Twitter account specially to communicate with @ATTCustomerCare (since they were ignoring/blocking my regular account), I finally found somebody who reached out and called me.
He too hit the problem that the order cannot be canceled, and he tweeted the error message to me:
That is a little cryptic, but WMS apparently means “Warehouse Management System” and he also added the comment “It’s suppose to go out on 9/19” followed by “Remember it’s a system that is calculating that but since you ordered day one I don’t forsee a problem.” That is certainly what the original date showed on the order, and I live in hope that the earlier comments I received yesterday were wrong and in fact it is the estimated delivery window in the status message that is inaccurate.
Meanwhile, I am going to compare the cost of T-Mobile for our five lines and see how that would work out.