I do not think the problem you are describing is related to the addition of the snubber. I think it is simply a matter of not biasing/ using the circuit correctly.
I assume the bipolar transistors were used just to simulate things, they are not part of the actual circuit. If they are, then remove them. The IR2110 is designed to drive the MOSFETs directly.
I do not understand why you use two switches on a motor, if you are not using a bridge configuration.
The high-side driver in the IR2110 uses a capacitor to act as a "flying" power source, connected between the source of the upper MOSFET and the Vb pin, which would be the collector of the NPN transistor in your drawings. This cap would allow the actual high-side driver to turn on the upper MOSFET.
BUT, this capacitor needs to be charged up periodically. That happens when the top MOSFET is off and the bottom one is on. Then the cap charges up from a low voltage source (say 12V), through a diode. As you can see, there should be a time when the bottom MOSFET is on and the top one is off. I don't think this happens in your circuit. Thus, the capacitor (I don't see it in your schematic, but I assume it exists), does not have time to charge up and the high-side driver does not work properly (it should stop, since it has an undervoltage lockout).
Now, when you only had the high-side drive, whenever the MOSFET was off, its source would go to GND potential, through the motor, and the cap could charge up correctly. I suspect with the addition of the bottom MOSFET that does not happen anymore, since I suspect you do not turn on JUST the bottom MOSFET for any period of time to allow the cap to charge up. You are forced to turn on both MOSFETs at the same time, which is not the way the IR2110 was intended to work. If it is used in a bridge, then the bottom and top MOSFET are not driven simultaneously (one bottom MOSFET is on and the top MOSFET in the second branch, controlled by THE OTHER IR2110 is on, then the roles reverse)
In conclusion, since you are not using a bridge configuration, remove one of the switches. I do not see what advantage you get from using both MOSFETs.
It makes a lot of sense to remove the high-side one and use the bottom MOSFET only, since it is easier to drive. (You may also find that the external diodes are not necessary, some MOSFET body diodes are rated for use in bridge motor drivers).
Or, use a bridge configuration, which will allow you to reverse the sense of rotation, too. However, the caps must be charged periodically in this configuration too, so you still need to have the bottom MOSFETs on periodically (and when the motor is stopped, to make sure the caps are always charged up properly). This is easier, though, since the top and bottom MOSFETs controlled by the same IR2110 are NEVER on at the same time, unlike in your circuit.
I hope this helps.