Adjusting the truss rod will not reduce the “rather high action”, it will simply reduce or increase the amount of neck relief (bow). If the action is too high and the nut is correctly set, you need to reduce the bridge/saddle height first. If it’s already as low as it can go, the neck may need to be re-set which is a job for a luthier and will potentially cost more than the guitar is worth.

To answer your question, if the truss rod pocket is wide, it may well have a nut rather than an allen key socket - take the cover off and have a look. I’ve never owned a Takamine so I wouldn’t know.