It's often better to supply separate power to high-current applications (such as motors) rather than taking power off the Arduino. This avoids problems with current and thermal limits of the Arduino on-board power regulator.
Looking at the manual for your board, the motors can be powered by a separate power supply hooked up directly to the motor shield. There's a six-pin header on the board marked VIN and PWRIN. When the jumpers are on the VIN side Arduino-supplied power is used to drive the motors; when the jumpers are on the PWRIN side the motors are supplied with power from the terminal block next to the jumper. (The rest of the board remains powered from the Arduino, but that's usually a negligible amount of current compared to your motors.) A photo in the manual demonstrates the pins set to PWRIN and an external power supply hooked up.
The documentation says you can supply any voltage between 4.8 V and 35 V DC, but be careful! What they don't make clear in the documentation (but is clear from the schematic and the L298N datasheet) is that this power is supplied directly to power side of the L298N and the motors without further regulation. The chip can handle up to 50 V DC, so it would be fine with a 35 V input, but you'll want to check that your motors can handle the voltage you want to give them.
If the motors can handle 19 V input, and your 19 V power supply is reasonably well regulated itself (which a laptop computer power supply generally would be), you can use that to power the motors directly through the motor shield and continue using your current power supply to power the Arduino.
(The ST L298 page has further information, including a slightly less readable copy of the data sheet and application notes.)