I don't know where the 350mA comes from but the motors can take over 1 amp on load: -
Specifications
voltage:
operating range: 1.5-3V
nominal: 3V
no load:
speed: 12511rpm
current: 0.29A
at max. efficiency:
speed: 10012rpm
current: 1.16A
torque: 15.7gcm
output: 1.61W
eff.: 42.36%
stall torque: 78.4gcm
On no-load the supplier states 290 mA but at max efficiency (somewhere about full load) the current is 1.16 amps.
This rules out using the SN754410. From a 5V motor supply, it will produce about 2V for the motor (due to gross inefficiencies in the transistor output stages) and, it will get too hot too quickly and fry. Any higher supply voltage and the device fries more quickly.
If you look at the full data sheet on page 4 you'll see that the absolute maximum power that is permissible to dissipate from the SN754410 is 2.075 watts. Given that the motor can take 1.16 A at or near full-load and the volt drops in the chip will be somewhere about 3V, the power dissipation will be nearly 3.5 watts just for one motor.
Note also other areas of the motor spec: The current of 1.16 A is at a torque of 15.7 gcm whereas the stall torque is 78.4 gcm - this means that if the motor stalled it is likely to demand a current of over 5 amps.
Here is a related question on the SN754410 that shows where on the data sheet the internal volt-drops come from. There are also recommendations for alternatives BUT you do need to state/know your motor supply voltage.