I am not familiar with that IC so I have no idea what its timing looks like.
To really understand the problems you are facing, you need to understand stray capacitance in a MOSFET. The simple model that everyone draws is deceptively simplistic. When you are dealing with small MOSFETs, you can get away with it. When you are dealing with the huge MOSFETs you have chosen to use here, the internal capacitance in the MOSFET can overwhelm the drive circuit you are using.
Lets look at your circuit dynamically.
When you applied logic 1 to the A input, what happened? Q6 and Q8 turned on, as did Q25. Q25 caused Q1 and Q3 to turn on HARD. That caused PAD4 to go from about 0 to about 12V FAST. Remember that point.
When the above was happening, what was going on with the B input? It was left unconnected and pulled to ground by R5 and R7, so no problem? WRONG. There is a very large parasitic capacitor between the gate and source of the MOSFET and another huge capacitor between gate and drain. Remember the fast voltage on PAD4? It couples to the capacitive divider formed by those two capacitors inside Q5 to momentarily pull the gate high. Is this voltage high enough to turn on the MOSFET? Sometimes yes! The same thing is happening with Q7. I do not have to point out the consequences if Q1, Q3, Q5 and Q7 are all on at the same time. The circuit becomes a smoke generator circuit
If the rate of rise of the voltage is fast enough and to a high enough voltage, the MOSFET will turn on briefly. This is called phantom turn-on and even many professional engineers are bitten by it. That is why real MOSFET drivers typically have 1A or higher drive capability and have R10 in the range of 1 ohm.
You can mirror the analysis if A was left floating and B was driven. You also have the same issue with the top MOSFETs when the bridge turns off, the flyback voltage drives one drain voltage high and then the bridge turns back on in the same configuration and the bottom MOSFETs turn on fast to slam the drain of the top MOSFET to ground.
The bigger you make the MOSFET, the worse problem you have with internal capacitance, both for phantom turn-on and for the RC delay due to the gate capacitance. Your main problem is that you have too much of a MOSFET for your driver circuit and we have not yet talked about Miller Capitance which is another parasitic effect that is not your friend. If you had to use these MOSFETs, you would
ABSOLUTELY have to use real MOSFET drivers that have the drive capability to fight this capacitance.
How to improve your design?
Why did you use such massive MOSFETs? If all you are doing is 6A, why have two MOSFETs in parallel that are collectively good to 100A?
If all you are driving is 12V, why are you using 60V MOSFETs?
I would start by SEVERELY scaling down the MOSFETs. Maybe four 20V MOSFETs that can handle 10A to 20A current each (no doubling up). I would stick with TO-220 and definitely heat-sink it to make sure it does not over-heat. Remember to insulate the tab!
I would seriously consider using a MOSFET driver chip, maybe a half bridge driver that handles the cross conduction prevention. Do a search on Digikey for "MOSFET DRIVER" The way things stand, you are not fully turning on the bottom MOSFETs. The IC is most probable providing only 3.3V, maybe as much as 5V which is still not enough to fully turn on the MOSFET.
Also, make sure that there is a
GOOD ground connection between the ground of the servo IC and this driver board. Two volt ground bounce and the 0V on the driver IC becomes 2V on the MOSFET driver and you have both A and B high and lots and lots of smoke filling the room.
Good luck!
---55p
Edited to add: Lest someone gets the wrong impression that this is a comprehensive discussion of MOSFET issues, this is a very superficial discussion. Bond wire inductance, Miller capacitance, lead frame inductance/capacitance, gate charge, transition times etc. all become important as you get into higher speeds and higher power. The "power" in the "high power" is what the MOSFET is rated for as well as what it is actually being use for.