That's true, you can boost the output current of 78XX regulators by adding an external power transistor .. see attached picture ..
Be aware that droping voltage from 19V to 5V means that the equivalent power of [19V-5V]*Current is converted to heat, so you have to mount power transistor and voltage regulator on a decent heatsink ..
Just Open your PS and adjust the biasing in the feedback circuit in the 431 IC. If require, just read the TL431 Datasheet from Internet. Then fix it at 12 Volt. Make 5 volt with any linear regulator like 7805 if it';s load requirement not too much (I Think). Adjusting 431 biasing is the best idea.