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 just (five days ago) made a date/time change on one of the flights on my current itinerary and retained the same ticket number, no reissue.
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.
Even better, below the "
Get Trip notifications>" shown above on the left, you'll find choices to change seats, cancel trip, add bags, and "
Show more". Click "Show more" and you'll get a drop down menu, -- one of the choices you'll see is "Print trip and receipt". That will give you the complete itinerary, ticket number, etc.