I have a slightly different opinion than some. First, the track alignment on my Z06 is -2.8 front and -1.5 rear. Second, the toe-in on the front is 1/8", and the rear is 1/16"
toe-out. As far as air pressure, I shoot for hot pressure of 32 lbs., depending on track conditions. On hot days I go out at 26 lbs. This may be your problem.
Regarding the NT-01s, I have run them on both the Z06 and a Boss 302. I will normally get 12-14 track days on the Z06, slightly less on the Boss. I have found that the tires heat cycle out and get hard before the tires cord.
Of course all of this depends on the track surface, the technical characteristics of the track and driver experience. Some tracks will eat through tires twice as fast.