Yes, the sould dies. The soul is just the person. The Bible even speaks of the souls of animals meaning the individual creatures.
Ezekiel 18:4 Behold, all souls are mine; as the soul of the father, so also the soul of the son is mine: the soul that sinneth, it shall die.
Romans 3:23 For all have sinned, and come short of the glory of God;
The logic is plain:
The soul that sins shall die
All souls sin
Therefore all souls die
In Genesis 2 we read that God created Adam out of the dust of the ground and breathed into his nostrils the breath of life and man became a living soul. What happens if God removes the breath of life? Man becomes a dead soul.
Job 34:14,15 If he set his heart upon man, if he gather unto himself his spirit and his breath; 15 All flesh shall perish together, and man shall turn again unto dust.
Here is a link to information you will find useful:
I hope you have found this helpful.
God bless you,