Attached are some IC that do gate drive like you want. That way the transformer can go away.
Some of this type of parts have a circuit that will not let both MOSFETs to be on.
You are using LTspice so look up Linear's parts. I know I have used a MOSFET Gate driver with LTspice. I just can't remember what I used. I will look it up in the morning.
On the transformer; use a resistor across the gates to dampen the ring. (as I said) or use a RC.