You really need to build that circuit with a MUCH better layout. You must eliminate those long wires between the drivers and the MOSFETs, and also all other long connections that carry high frequency signals.
A suggested configuration is this: Make two groups of components. Each group contains the two MOSFETs of one half bridge, the driver chip driving them, and the components that go in between. The two MOSFETs are directly together, almost touching each other, and you place at least one nice, very low ESR, very low inductance bypass capacitor across them, from the drain of the upper to the source of the lower MOSFET. Those connections are critical, and should not be longer than a few millimeters, and as wide as possible. So use a circuit board with copper planes for those connections, not narrow tracks.
Those two groups are then arranged on the board so that the sources of the lower MOSFETs end up as close together as possible. This is necessary because your non-isolated driver chips can only tolerate about 5V between those MOSFETs sources and the input-side ground to the chips, and to keep the switching transients there smaller than 5V you really need the negative ends of the half bridges very close together, and very close to the driver chips.
The connections between the driver chips and the MOSFET gates, and the Vs connections to the sources of the upper MOSFETs, are slightly less critical, but not much. Keep them shorter than 5cm total, and if you can, much shorter. Also it's good to keep the gate connections and the returns from the sources to the driver chips, and their bypass capacitors, very close together, either using side-by-side tracks on the board, or better using tracks on both sides of a double sided board, running one atop of the other for as much as possible of the distance.
The layout you used, with long wires between the MOSFETs, and MOSFETs and their drivers, is totally forbidden when working with high frequency! You can use that in linear power supplies, and maybe still in audio amplifiers, but definitely not in a circuit hard-switching high current at 100kHz! To get nice clean switching, you need to cleanly reproduce a lot of harmonics of those 100kHz, and for that you need a flat bandwidth to about 10MHz.
The strong ringing your oscillograms show is generated when the driver chips excite the resonant circuits formed by the MOSFETs' capacitances and your long wires' inductances. You cannot eliminate the MOSFET capacitances, but you can sharply reduce the wiring inductances by following the advice above. That will move the resonances to a much higher frequency, and at that frequency the reactances of the MOSFET capacitances is much lower, so that it can be better damped by the gate resistors, by the RDSon of the MOSFETs, and by stray resistance.
If after all that you still get too much ringing, you can add an RC snubber at each half bridge output. You can choose the resistance of that snubber such that at the full high voltage it takes roughly the same current as the inverter's maximum load, and select a capacitance that has about the same reactance as the resistance value at the ringing frequency. This simplistic method works well enough as long as the ringing frequency is MUCH higher than your switching frequency (say, something like 100 times higher). If it isn't, then you should first bring it up there, by reducing the wiring inductance, bypass ESL, and if that's not enough, chosing FETs having lower capacitances, or as a last resort, reduce the switching frequency.
Another option, elegant but more difficult to design, is to absorb the wiring inductance into the design of a resonant converter.