How do I remove the feedback noise in my circuit?
Two problems. If you took away the inductor you are charging a capacitor and then discharging it. That  is just 100% wasted energy given off in heat. The second problem with your H bridge is that the high side N channel MOSFETs are source followers and will never switch on to a low on resistance: -



Shoot through might also be a big problem. Gate charge and discharge times due to the gate-source capacitance will also mean inefficient switching and this will burn energy. A lot of MOSFET drivers are capable of dumping about an amp into the gate to turn it on sub 100 ns; a regular flip=flop probably can't produce more than 20 mA.

Efficient H bridge design is never accomplished with a simple circuit. Try looking up MOSFET ZVS circuits for an increased efficiency: -



There are only two MOSFETs and the power feed to the coil (via the inductor at the top of the circuit) allows the coil and capacitor to freely resonate and hence efficiency is much better. If you don't like the split centre tapped coil you can use two inductor feeds from the positive supply to each MOSFET drain.