Flyback Transformer Troubleshooting
At 355kHz and (let's say) 50:50 duty cycle, the MOSFET will connect the transformer primary to 24V for a period of 1.41 us. The data sheet linked covers inductors from 10uH to 25uH so, assuming 25uH, the question is "what has the current in the primary risen to after 1.41us has elapsed": -

V = \$L\dfrac{di}{dt}\$ so \$di = V.\dfrac{dt}{L}\$ = 24 x 1.41us / 25uH = 1.354A

The data sheet says the maximum primary current of the 25uH model is 1.4 amps so, if your duty cycle goes a little above 50:50, then you are exceeding the primary current rating and the core will start to badly saturate and the device will certainly fail after a short period because saturation causes more current to be taken.

If it's a 10uH primary inductance then the peak current will be 2.5 x higher at 3.384 amps and both of the two 10uH transformers in the data sheet will be seriously affected because the best one has a max primary current of 1.4 amps.

However, the LT3420 will come to the rescue to a certain extent - it nominally current limits the primary current to 1.4 amps (20mV internal reference and 0.015 ohm internal measurement resistor) but, it could be a maximum of 1.6 amps and at this level I feel your design is flawed.

I feel you are asking too much of this transformer.