Will this circuit suffer from Shoot-Through? and is there a quick fix?
As a solenoid driver, the duty cycle is likely to be so low that while shoot-through will happen, the effects might not be serious enough to worry about,(especially given the 3V supply shown). 

You would have to satisfy yourself that the power dissipaton in that 20ns, times the switching rate, is within acceptable limits; so that for instance it does not result in an unacceptable temperature rise in the FETs or exceed their max current ratings. For a solenoid the switching rate is unlikely to be more than a few Hz.

If it is, then adding complexity to the design to overcome shoot-through is unnecessary (though it may still be desirable, if it affects battery life for example, or puts unacceptable current spikes on the supply rail).

However, anyone taking this power stage and re-using it for a fast switching PWM circuit had better watch out!