Regulator Getting Hot. Is The Mosfet Driver At Fault?
If your total consumption is 1.2A you  are dissipating 8.4W which is a challenge even with a heatsink and fan.  If your thermal impedance is as good as 10C/W your temperature rise would be 84C over ambient.  This is a good use case for a buck regulator rather than a linear regulator.
Your FET driver should consume minimal average current unless you're switching the load very fast.
You don't say what your gate drive voltage is, but if it's 5V then the power consumed by driving the gate is P=QVf where Q is the total gate charge at 5V, V is your 5V gate drive, and f is the switching frequency.  So power consumed is $$98nC*5V*38kHz = about 19mW$$
There's another couple of mA for the quiescent current drawn by the driver chip, so in total not much at all.