Crossover distortion returns when load is applied
That's not completely unreasonable. Your distortion seems to last for about 3 usec. 

First, your bias voltage (3 diode drops) is about 2 volts. Note that the data sheet give 2 volts as the minimum gate threshold, and it could be as much as 4. At high load impedances the FETs are close enough to turn-on that you don't see anything, but that's not enough at lower impedances.

An obvious thing to try is to add a couple more diodes. This will possibly have a serious drawback - excess power dissipation. YMMV.

Another issue to consider is the op amp. Your unit would seem to have adequate speed and slew rate, but remember that it's feeding a fairly large (~4000 pf) combined MOSFET gate capacitance, so it may be struggling to keep up at the zero transition.

I suggest you rerun your scope measurements, but this time also look at the op amp output voltage. You may be surprised by what you see.