IPhone randomly turning off

Might be a battery calibration aka battery hardware to software communication issue where the iPhone's thinks there is more power than the battery can provide. When was the last time (before the iPhone started to just shut down at 19% battery) that you did a battery calibration as recommended by Apple.

Use iPhone Regularly

For proper maintenance of a lithium-based battery, it’s important to keep the electrons in it moving occasionally. Be sure to go through at least one charge cycle per month (charging the battery to 100% and then completely running it down).

If you don't then what you are describing above may happen where the phone just shuts down with a random non 0% battery percentage left. If the problem still persists after doing a couple of battery calibrations then it may be that the battery or other hardware is failing, and that the iPhone should be taken to a Authorized Services provider / Apple for more attention.

It sounds like you need to backup your iPhone with iTunes and change your battery... but I'm a little confused by your post. You have tried to put a new battery in it already? If so your issue is probably related to other internal components of your phone and not the battery. The 3GS is quite old now. It may be time for an upgrade.