It turns out when I ran the Carduino at about 15v, it would be stuck in a reboot loop, and it was because of the regulator. I decided to just swap it with a LM317 (it is good up to 28V, why not?). Well, it got hot, extremely hot (linear regulating 16v to 3.3v at 60mA is a good deal of power that needs to be dissipated).
Since I just needed 3.3V, I decided to just drop the voltage using the 1N4001 diodes, 5 of them. And it works, so I’m happy 🙂