Obviously they're current limiting for motor stalls I would imagine. I'd leave them there. I think it's prudent to replace the motor as well as the board, you're already paying shipping, so the motor won't cost much more.
Did you think to measure the current drawn by the motor? Did you try applying a light load by gently holding one of the wheels to provide a load to the motor? The motor should not draw more than around 50ma at 6 VDC free running. When I applied a generous amount of friction to the shaft, the current was around 100ma. I've seen them draw much more when defective, that causes them to kill the driver transistors on the reverse board.
