Inverter Design - High Side MOSFET Switch-off Issue
It looks like a straight drive problem to me.  The gate and drain curves have a relationship that looks reasonable, so I think you are getting a good measurement.  The only thing to do is to decrease the drive impedance.  You can reduce the inductance in the gate drive lines by making the path as short and fat as possible; remove the coupling to the current path by keeping the gate drive path from laying on top of the power and ground planes; but to really improve you will have to get a more muscular driver to overcome the negative feedback from the gate-drain capacitance.  This symptom will become more pronounced with the higher voltage because the higher dv/dt will mean more voltage on the gate.  You can shop for a FET with low gate capacitances, but these generally have higher "on" resistance.

As I said earlier, this is not a terrible switching waveform; you can probably run a motor with moderate current but you will have losses that will heat up the FET.  However if you are running high currents, you have probably noticed that "real" gate driver IC's often sport output currents in the 5 ampere range.

One other thing you must do in any case is to provide a big low ESR (ceramic not electrolytic) decoupling cap for your drive circuitry.  There must be an adequate charge reservoir to provide that instantaneous ampere or two that you will need to overcome the dv/dt coupling effect.  I can't tell if you have a 2.2 uF ceramic from your schematic, but I'm guessing you do.  If you do put in FET drivers, the capacitance is necessary.