Have you used PWM to control the motors' speed? You will need that for the higher voltage supply.
The 1A supply cannot handle both motors' stall current of 1.6A combined, assuming the motor shield
is taking about 4V drop (no unusual for L298 motor shields). Thus with the 1A supply the motors simply
pull its voltage down to a few volts on start up and thus USB power is needed for the Arduino.
That's normal - I wouldn't expect to power the Arduino from the same supply as the motors in the first
place.
Yeah I just had it set to full power just to test out the power supply, and that's when it absolutely takes off. Just want to make sure it's not too much power, but it sounds like you're saying the power supply I'm using can't even reach the max?
Don't supply more voltage to the motors than they are rated for (unless you are compensating in PWM control) and they will take only the current they need. If you make sure there is a big capacitor in parallel with the controller it will give some kick for start up, if you want to get fancy start on low % of PWM and work your way up.