Do I need a resistor or no?
LEDs are diodes, so over the normal operating range the voltage is relatively stable for wide ranges in current.  However, this also means that if you were to drive a LED with voltage, small changes in that voltage will cause large changes in the current.  The characteristics are not definite enough to make this current predictable.  LEDs therefore need to be driven with a controlled current, not with a low impedance supply.

Your basic flaw is that you are trying to drive too many LEDs in series for the voltage you have.  Get a higher voltage supply, lower voltage LEDs, or put fewer of them in each series string.  With 5 LEDs in a series string, the nominal drop will be 15 V.  That leaves 4 V for a resistor to drop, which is good since that will reduce current variations as the LED voltages vary.  4 V / 20 mA = 200 &Omega;, which is the size resistor you should use with 5 LEDs in series driven from a 19 V supply.