With CX flights, I try now to only book them on the AS website. I don't consult other websites first, as I'm typically looking for an enroute stopover (i.e. in HKG) --- which, iirc, BA/AA/QF websites don't support. I find that piecing trips together myself is vastly more time efficient and likely to succeed than speaking to a friendly (but frequently clueless) AS telephone agent. That said, once I receive the email confirmation, any/all subsequent issues get addressed to telephone customer service.

[One issue that I encountered last week was that a confirmed, ticketed CX reservation suddenly disappeared from the app. I called and spoke to someone who, in their infinite wisdom, decided that the cause was that we had not in fact been charged the $12.50 partner fee, notwithstanding that it was printed on the email receipt. She began muttering and fussing, and finally sighed when she saw that it was now again visible in my account. I thanked her, and hung up, and subsequently received a new confirmation receipt, indicating that I had now paid $25 in partner airline fees for this one-way trip! I'll wait a few weeks/months for all the debits and credits to post to the credit card, and will take it up again if we've really been double-billed....]