How should I modify this PCB antenna?
The results you have are a bit odd, due to the lack of a ground plane under the antenna (as the datasheet for the antenna suggests http://www.ti.com/lit/an/swru120c/swru120c.pdf) the PCB thickness should not matter, there isnt a reference plane that will modify the antenna characteristics themselves (you did remove the ground plane from underneath the antenna right?)

What does change however is the feed line from your tranciever to the antenna, which will need to be modified to maintain the same impedance as before, as your layer stackup is different.

Also, as SteelRat pointed out in the comments, you might want to use the same copper thickness as the TI stackup, as that has some effect on impedance, he indicated it is the copper thickness that might make or break the antenna. So if you follow TIs recommendations with copper thickness, shape, ground plane recommendations, and modify your feed trace to 50 ohms, then your design is quite likely to be good and stable.