Hi Grayson,
One more thing to look at, in addition to the stability. What is the connection to the power supply like? If long leads are used a bulk cap may help on the input to the board. On the second screenshot there is a lot of movement VIN when in the unstable region. Are there oscillations on VIN with the higher input voltage due to the lead inductance? From the description of the failed part, with VIN shorted to GND, it makes me think over-voltage occurred on the VIN pin.
However it appears stability is the main issue due to the very low ESR output capacitors.
Best Regards,
Anthony