Though, the emitter is pulsed and I also rely on the fact that it stops resonating after I cut out the power. Unless my LC oscillator has a low Q, it will keep oscillating after I cut out the supply, which is something I'd prefer to avoid.
Correct! You can add a resistor in parallel to reduce Q (but it will naturally "not be perfect" since the transducer is putting energy out as sound waves and it's not really a capacitor.