So, instead of regulating voltage down to 6V / 3A and trying to drive the guns with that, I switched my driver board to use the full battery voltage, and instead control it with PWM.

Unfortunately, this made the N-channel low-end switches fail "short" -- as soon as I put on voltage, the guns fire like crazy.

I was wondering why this was, but then I consider the circuit: Voltage, inductor with capacitors, switch to ground.
That looks a whole lot like the first stage of a boost converter!
With 16V battery, and about a 40% PWM duty cycle, it's totally possible that I exceeded the 30V my N-channel MOSFETs are rated at, and thus I may very well have over-volted them and blown them.

To verify this theory: Is the common failure mode for over-volted N-channel MOSFETs to fail "short" (stay open)?

I've only ever failed MOSFETs by overheating them and having them crack in home-brew H-bridges before.