Opting out of ATT seems like the most obvious no-brainer, but are there options for phone service that are actually halfway decent?
Or as an alternative, would it be feasible to get some kind of internet phone/texting service, use that over wifi, or maybe have a basic data plan to go with it at most?
Go for a prepaid provider and give out as little personal information as necessary. Avoid the major carriers directly because they need a social security number.
That said, mobile phones are inherently not private. No matter what provider you choose, they will be able to track your location using tower triangulation. Even if you give a fake address, it would be pretty easy to identify you if you always have your phone on at home.