Oh, I think I see what's going on now.
It's a constant guessing game since no setups are specified for each waveform here, but I think the last waveforms must've been scope probe on secondary, unloaded (no gate connected), primary drive as shown (TL494 + UCC drivers). So, you get the response of an unterminated transmission line, and apparently the winding is about that long (10-20ns electrical length, so, about 2m of wire?), so that's that.
This waveform is nothing of concern: the gate resistance swamps transmission line behavior, and leakage inductance dominates.
2m of wire at ballpark 100 ohms Zo is 0.7uH. The gate average capacitance is 20nF per transistor, so the LC series resonant circuit has Zo (that's a different Zo now) 5.9 ohms, so the 6.8 ohm gate resistor will do well enough, and it'll have a time constant around 180ns. We can expect about 1/3 that or 60ns commutation speed at the drain. (Actual values may differ, but will likely be within 2x or so of these.)
To run faster, smaller gate resistors are not feasible: that'll just result in ringing. A lower transmission line Zo is required to enable that. That is, use more twisted pairs in parallel, or star quad(s).
The slanted waveforms shown earlier are suggestive of excessive magnetizing inductance; rewind with more turns or a higher A_L core.
Also consider an R+C in parallel with the GDT coupling capacitor, to dampen startup transients. Probably there's some low frequency ringing in there. Same calculation, R = sqrt(L/C), where L is magnetizing inductance and C is the coupling cap; use a series capacitor >=2.5 times the coupling cap.
Tim