Prod ATT to Open an Engineering Ticket?

I've blown 2+ hours on three occasions with ATT's offshore "support" who have told me to wipe my SIM card with alcohol, make a call from a competitor's tower, pray to a god with six arms (OK, not exactly but pretty close.)

I can make a call from every tower but my home tower.  Everyone else on our plan can make calls. I've swapped my SIM into three different phones, and still get the "Network not available" message.

No problem with this very strong signal tower, until two weeks ago.  And only my account, regardless of phone.

How do you get ATT to open a ticket with local engineering for them to clear this tower's routing in order to stop rejecting my calls?

We all have unlocked, multi-band phones, so my next stop is Verizon.

Thanks for any insights.