The 2N6782 N-channel FET is rated at 100v drain to source at a maximum 3.5A and 15W max power. I picked a through-hole part because the surface-mount part would be difficult to solder by hand.
To drive the FET, you can use the LT4440-5, which takes a logic level in and can drive an N-channel MOSFET switching up to 60 or 80V depending on the version. The supply current is negligible, only a few µA. The driver only comes in a surface mount part (SOT 23-6 or MSOP).
The only drawback is it requires a supply voltage (VCC) between 5 and 15V (thanks to the OP for pointing out this variant of the LTC4440 with a lower minimum supply voltage). Since you have only a 3.3v supply, you could use either a voltage doubler or a simple boost converter (such as the MIC2141) to supply the VCC voltage.
Both the driver and voltage doubler/boost converter should only require a few ma (if that) from your 3.3v rail.
The FET and driver should work up to 1 MHz, maybe a little higher.
You're only looking at Vg on the top fet, not Vgs, which is what actually turns it on. Unless you know something which is not on that scope plot, then that first rising edge is not the top FET turning on, it's the source and gate rising up as the bottom FET turns off.
This is what a typical high-side MOSFET driver does in a half-bridge - it floats the drive of the top FET on top of the upper/lower source/drain junction node.
Update:
OK, assuming that what I already wrote is not the confusion, here's a mechanism which can cause brief shoot-through on half-bridges with big FETs, though it happens at the top turn-on point, not the bottom turn off:
A MOSFET has an implicit capacitor between gate and source, which everyone knows about and is what you have to charge up to turn on the device. However, it also has capacitance between drain and gate. When the upper FET turns on, and pulls up its source, current flows through this capacitance in the lower FET into the lower FET's gate circuit. Depending on how hard the lower gate driver and lower gate resistance hold the gate down, you might see it turn on briefly/slightly as its drain, and hence gate, is pulled up by the top FET.
This tends to be a much worse problem when the PSU is very lightly loaded.
The problem with just changing both gate resistors is that you don't see much difference, because although slowing-down the turn-on of the top FET helps, increasing the gate resistance on the bottom FET makes the problem worse, and the two effects cancel-out.
To start with, you could just slow-down the top fet by increasing its gate resistor. You may then have to increase your anti-shoot-through delay to avoid problems in at the other end of the cycle, but I'd suggest you make that absurdly big at the moment anyway to eliminate that from your search.
I have ended-up with diodes around gate resistors in this situation, to allow me to set turn-on and turn-off rates separately.
You might find looking closely at both ends of your bottom gate resistor with the scope lets you see this happening. It doesn't sound like you have a current probe available, but for the sake of experiment you could perhaps try adding some low resistance in the 0V line somewhere which would let you see exactly when the shoot-through current spike occurs.
You should post the schematic.
Best Answer
Driving the gate to 3.3 V is within specification, so nothing bad should happen.
However, note that Rdson is only guaranteed for 5 V gate drive. The graph of Rdson as a function of gate voltage gives you some guidance what you will get at 3.3 V, but this is not a guarantee. It looks like you should expect around 600 mΩ. You can't base a volume design on "probably around".
Despite the marketing hype at the beginning of the datasheet about exceptionally low Rdson, the Rdson is actually quite high. The gate leakage current is also quite high.
It seems the unusual aspect of this FET is that it can switch 100 V with only 5 V gate drive. If you only need to switch 30 V, or can provide 10-12 V gate drive, there are much better FETs. This is definitely a specialty part. I haven't looked up the price, but I expect that says "specialty" too.
The package also pretty much requires hot air soldering for manual work, unlike a SOT-23 which can be soldered with a ordinary soldering iron.