DH and I have each had the Amazon card for years, and Prime membership off and on. Chase automatically switches between 3 and 5% for Amazon purchases, but as for the cash back, I usually just check every month when the statement drops and transfer the money then.
Oddly, my card (but not his) was recently targeted for 4% back on everything else (still 5% on Amazon and Whole Foods) for 6 months. You'd better believe everything's going on that card now...