You CAN drive a motor with a dual-MOSFET driver (some drivers even list that in their datasheet), but you have to realize that mosfet drivers don't have some of the features used in dedicated H bridge chips.
However, if it gets really hot, it might mean you have just killed it. The MOSFET drivers don't have as much protection from flyback as H bridge drivers, so using a high current motor will quickly kill them.
The driver you have can only dissapate 500mW of power, which is about .1A @ 5V, you won't be able to drive any big motor with that.
I am using a 9A dual-mosfet driver to drive the smaller steering motor on my robot (.5A stall), since my current consumption is low it works fine. But if I use them to drive my main motos (7A stall) they quickly get really hot and stop working.
The motor shouldn't spin if both side is low (or both side is high), rather it should brake the motor.
You can stack them in parallel to double the current rating, but make sure you have good protection diodes.