There is a way to tell if she spawned. Just go to x:34 y:13 (north of namai, next to a broken wooden shrine) after doing the Outfoxed FATE, wait for an hour (the spawn time window) and if the namai elder pops, the Foxy Lady fate is active out there, if an hour has passed. Rinse and repeat. That way you aren't running around looking for nothing.
In the same location, a small fox kit can be seen during the Outfoxed FATE.