On previous RTWs I have had a lot of rejected mileage claims even when I have appealed with my ticket number and PNR.
I've now discovered that AA is issuing new ticket numbers after each minor change, eg an airline changes a flight time or I change a flight date. I've been assuming that the ticket numbers initially sent to me are valid unless I made an actual routing change.
Additionally, AA sometimes only sends ticket numbers for one of the passengers in the email generated after a change. No one knows why. And the call centre can't email it you either ("we have had that facility taken away").
So my tip is that once your journey has commenced, after each (even minor) change, record the ticket numbers you are issued with, and the dates. That way you will be able to find out later which ticket number was used for a given segment.
On the AA website you can get them from this rather obscure place within the "Manage Trips" window (they aren't listed in the regular "manage booking" screens, nor in the "email me" email). Click on the little "I" and a window pops up with your latest ticket numbers.