AA has SWUs, but they're typically only handed out to EXPs. These are "instant" and are processed immediately if seats are available for this (dependent on revenue controls).

Here's more info on AA's 500-mile upgrades. Keep in mind that 500-mile upgrades do not clear until days or hours before the flight.

Sounds like your options are to use 500-mile upgrades (either ones you've already earned or are willing to buy), or go for the miles+cash option you eluded to earlier.