Thats what leakage inductance together with the load capacitance does: rings and oscillates. Your transformer might not be wound "tightly enough" with low enough leakage inductance. Also specifying a transformer only by number of turns doesn't really say much.
What is important, is to calculate the flux density in the transformer. It is not acceptable, to just take a random ferrite core, put there some turns and think it will work. It might and it might just not.
Usually, there is more into it: not only must be the operating flux density kept within a safe level, but also a magnetizing current should be checked. Mostly the transformer can work with a very low turn counts even with safe level of flux density, but such low number of turns implies low magnetizing inductance, thus high magnetizing current. That current is what bothers the gatedriver (integrated or transistorized).
For example if it is enough to use only a few hundred miliamps to drive the mosfets directly, it does not make sense to design the transformer with 2 amps peak magnetizing current. (remember, the driver has to deal with the magnetizing current too).
But that was just a side note about the transformer design.
The magnetic flux density can be calculated using standard formulas, like the Farady's law U = -dPhi/dt and the magnetizing peak current can be easily obtained from the differential formula of voltage on an inductor: U = L * di/dt.
Note you have to design in in a way the transformer will work correctly in the whole range: You have to count with the highest permissible supply voltage, lowest possible drive frequency, maximum/full duty cycle, etc.
To dampen the ringing on the outputs, I would recommend discarding the resistor from the primary completely (it shouldn't be there anyway if the driver does provide correct dead time generation which here it does) and putting it on each of the secondary. That will dampen the LC circuit (leakage inductance with the load capacitance).
Small amount of ringing can be tolerated on the waveform, but it have to be ensured, none of those peaks can switch any of the mosfets on any time. So the peaks on each gate have to be safely lower, than the mosfet gate threshold voltage, which might be as low as 2V.
PS: You can't simulate a mosfet gate with a plain capacitor. The mosfet gate is a nonlinear capacitor and all sorts of problems (like the miller capacitance Cdg must be taken in account).
PPS: You can also try to twist the wires together and then wind the toroid with it, if you haven't done it that way. That will also lower the leakage. You can also measure the leakage. Short one of the windings and measure the transformer inductance on another free winding. The uH you will see is the leakage mostly. The lower value measured, the better. A coupling coefficient can be also calculated from the ratio of magnetizing and leakage inductance.
P3S: The schottky diodes shouldn't be neccessary for the TC4420, as the driver has mosfet output stage with their natural substrate diodes. The schottky can only help lower the forward drop.