Let's back up to the beginning and your original circuit. As I suspected, all of your power rails share a common "ground" connection, i.e., the center-tap of the power transformer. This will not work for reasons already discussed!
The original problem you discovered with this circuit was the MOSFETs both got very hot. The reason for this was the low-side MOSFET source was tied to the -32 V rail. This effectively biased the gate of this MOSFET positive by 32 V and the positive pulses coming out of pin 5 (LO) just added to that. The result was the low-side MOSFET was turned on all the time. The high-side MOSFET also has positive-going pulses applied to its gate whenever the low-side MOSFET is turned off. Problem is, the low-side MOSFET never turns off! This places the -32 V rail on the source terminal of the high-side MOSFET too, effectively biasing the high-side MOSFET gate to +32 V (with respect its source terminal) plus whatever amplitude the positive pulse is. The result was the high-side MOSFET was also turned on all the time. With both MOSFETs turned on simultaneously, this creates a "dead short" across the +32 V to -32 V rails. No wonder the MOSFETs got hot!
The only thing that might have saved the MOSFETs from self-destruction is the Rds(on) value is specified to be only 1.7 mΩ with 10 V gate-to-source drive and 195 A drain current! That's 64 watts dissipation. Rugged little beastie. Your little power supply transformer apparently is not capable of providing enough current to destroy the MOSFET when the MOSFET is fully conducting. But that doesn't mean you are home free. The maximum gate-to-source voltage is specified to be only ±20 V before you punch through the oxide layer and destroy the gate. You are hammering it with 32 V! You may get away with that a few times, but it is not good engineering practice. The positive pulses applied by the IRS2153D to the MOSFET gates with respect to their source terminals should be the same amplitude as the zener voltage, about 15 V.
So, how can you fix this problem? Well, first the low-side MOSFET source must be connected to pin 4 (COM). This is true no matter how many other bells and whistles and circuit modifications you make from now on. For now, DO NOT CONNECT -32 V rail to ANYTHING! The original circuit should now work with a peak-to-peak output of 32 V at the connections between the low-side MOSFET drain, the high-side MOSFET source, and the load. The other side of the load connects to the low-side MOSFET source, which is now back to being at ground potential, per your original circuit with the -32 V rail not being used.
But, wait, Hop! I want to drive the load with bi-polar ±32 V! I want to turn the IRS2153D on and off with a transistor controlled by my MCU! I want to vary the frequency with a digital potentiometer replacing the 6.8 kΩ timing resistor! I want my ground, too! Well, you can have most of those things, but the -32 V rail has to be at ground potential. The only way I know to do that is to float the ±32 V power supply so connecting the -32 V rail to common doesn't short out the rail.
If you want bi-polar power applied to the load, one side of the load would connect to the "common" of the ±32 V supply. The transformer, full-wave bridge rectifier, filter capacitors, and ±32 V voltage regulator circuits must not share any connections with the rest of your circuit, and most especially the "common" for this power supply must not connect to the "common" of the IRS2153D circuitry. The only other thing that shares the transformer center-tap "common" is your induction heater load. Since the -32V rail is connected to the "common" of the rest of the circuit, the center-tap on the transformer will be 32 V positive with respect to that common. You would need to make sure the transformer insulation can handle that. When the low-side MOSFET is "on" that will apply -32 V to the load with respect to the transformer center tap. When the high-side MOSFET is "on" that will apply +32 V to the load with respect to the transformer center tap. You must not "ground" the center-tap of the transformer because that would short out the -32 V rail. This means you either need two transformers, or a transformer with two secondary windings: one center-tapped winding for the ±32 V supply, and another lower-voltage winding for the low-voltage power supply.
Floating is a simple concept: there must be no electrical continuity between the "floating" circuit and earth ground. Imagine the circuit placed in a balloon floating high above the Earth. Easier said than done because of leakage resistance and leakage capacitance, but usually a transformer provides enough isolation between the power-line ground and the secondary winding(s) to allow circuits attached to the secondary winding(s) to be floated up to several kilohertz and several hundred volts before the parasitic capacitance between primary and secondary windings becomes a significant factor.
When using your Rigol 'scope to measure the high-side MOSFET gate pulses, you can "float" the scope with an isolation transformer or (if so equipped) operate it from internal batteries. This would allow you to connect the probe common to pin 6 (VS) and the probe tip to pin 7 (HO) to measure the gate pulse applied to the high-side MOSFET. Bear in mind that this "floats" the entire 'scope, so whatever potential (with respect to "ground") the common lead of the probe is at will also be on the chassis and controls of the 'scope. Make sure there is no way the 'scope chassis can come in contact with a real "earth ground" while you are floating the 'scope to take a measurement. And, also, if you "float" one probe common you have floated both probes' commons. The two probe commons are connected together inside the 'scope, and probably connected also to the power-line (green wire) ground inside the scope.
Some daring souls will just use a "cheater" plug to disable that green-wire ground, relying on the 'scope's internal transformer for isolation from the power-line ground. But who knows what's inside a 'scope these days? An external isolation transformer is a safer method. If it comes equipped with a 3-prong grounding socket, with the green-wire grounding wire connected to the green-wire in the power cord, you may have to use a "cheater" plug to defeat the power-line ground connection.
Your 'scope traces look good. The little "glitches" at the square-wave transitions are there because the gate drive signals do not overlap. You can see this if you trigger the 'scope using the rising edge of the low-side gate signal on one channel and display the high-side gate signal (with AC coupling to remove the DC offset) simultaneously on the other channel.
Is it all clear now? Are you ready to re-connect the MOSFETs?
Hop