First, that's not a good application for a protoboard. There are just too many parasitic inductances, capacitances, and resistances. Use dead-bug construction.
Power bridges like this are actually quite difficult to get right and 90% of it is getting the gate drive right.
As a starting point, limit the current through the MOSFETs by adding a resistor in series with the drains of the two top switches. Likewise, for now, replace the load coil with a resistor. Then, when there is a problem, you don't blow everything up. Once you have the gate drive sorted you can take out the drain resistors out. Once you have the desired waveform appearing across the load resistor, you can change it to a coil.
The gate driver chip inserts a microsecond of dead time. But you can still get shoot-through if the MOSFETs take too long to switch. While you have the drain resistors in place, measure the voltage across them to check the shoot-through current. Smaller gate resistors will give you faster switching, but only to a point because the IC has a limited output current.
That's a 60 volt MOSFET which seems like plenty of overhead for a 12 volt system. But with an inductor in the load, you could generate more than 60 volts. Especially on a protoboard. (I once had a high-power bridge develop so much voltage across a 100 mm piece of 00 copper wire that it shot sparks across a couple cm gap.)
Unfortunately, you have a tradeoff here. Faster turn on and turn off will cause bigger voltage spikes from the inductances in the circuit (especially the protoboard). But slower turn-on and turn-off result in higher power dissipation in the transistors. Higher edge rates also make EMC problems more severe but you probably don't care about that at this point.
As someone else said, driving the pictured coil with 12 volts with a 35 KHz square wave is a non-starter. You will either need a higher inductance coil, or a much higher drive frequency. Protoboards are useless at the frequencies it would take to drive such a coil with 12 volts. What are you actually trying to achieve?