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"Do I have to first transfer my points to a given airline like American and THEN search through American for a free flight? Or can I search for availability on a variety of airlines and then transfer the points to that airline?"

Anyone -- even those without an AAdvantage account -- can search for award availability on AA metal at aa.com. If you have an AAdvantage account and see availability, you can place the award "on hold" for five days, and then initiate a transfer of SPG points to AAdvantage. Unfortunately, such transfers might not always occur within five days of requesting one.

If your award travel using AA miles involves even one flight on a partner carrier, you would have to call AA to search for and book such an award (unless you wanted to use some other online engine to search for award availability on AA partners); AA would again hold your award reservation for five days.