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This is the "classic" waveform of a push-pull-converter with an iron core transformer without a reset winding.
I know....I built several just like that in the late 1970s utilizing a 3525.
No simple solution to clamp the narrower spike. There is a lot of energy in an iron core magnetizing inductance.
And I am sure that your transformer is also overheating.
My advice: Go for the high frequency, "modified sinewave" route.
The contactor shakes because you are not supplying an AC waveform with a low impedance, when the switches of your inverter turn off the current in the contactors drops to zero quickly and they start to release, if you put a 200W light bulb across the inverter output - this will help as it lowers the impedance of the AC supply - less shaking in your contactors ...