A reservation doesn't mean the ticket has been issued. Even on AA I can reserve, select my seats, and pay but the ticket might be hours or days later.

For paid travel I use a travel agent where it's done in seconds by them.

All said it doesn't mean you don't have a proper reservation. The e-ticket number would be the same all the way so I bet you can't see a ticket number for the AA flight. If you can, that's the ticket number.

Try the locator code, six letters.