Mormons believe the soul/spirit is immortal/eternal. There are four phases of the soul:
- Pre-earth life-soul before birth
- Earth life-soul joined with physical body
- Spirit world-after death soul goes to spirit paradise/prison
- Resurrection-soul is reunited with perfect body forever
Onto the scriptures. Acts 3:23 has a footnote on the word destroyed that points to two scriptures in Doctrine and Covenants, both saw basically the same thing so in D&C 133:63
63 And upon them that hearken not to the voice of the Lord shall be fulfilled that which was written by the prophet Moses, that they should be cut off from among the people.
So destroyed in this context just means to be cut off from among God's people, or the saints.
Ezekiel 18:4 mentions the soul dying. Mormons don't believe death means destroying or the end, rather it means separation. Physical death is the separation of spirit from body, but it isn't this definition used in this scripture. The other death is spiritual death, where when man sins he is separated from God's presence, which is the usage in Ezekiel 18:4. If a man repents he forsakes sin and turns back to God, and thus can overcome spiritual death.
Psalms 146:3-4 is referencing physical death and once the spirit leaves the body the body returns to the earth. The body no longer has a spirit and has no thought, the spirit still lives in the spirit world. James 2:26 on the body being dead:
26 For as the body without the spirit is dead, so faith without works is dead also.
Also in Psalms 146 His and he refers to the physical body since it mentions
he returneth to the earth; so the next line refers to the body losing it's thoughts, because the spirit has left it, not the spirit dying (being destroyed/ending).
From Answers to Gospel Questions Volume 4 page 183 by Joseph Fielding Smith (the question that is being asked is on Ecclesiastes 9:4,5,10 and seems like a similar question to yours):
...the body is laid away in the grave peacefully and it knows nothing as to the affairs of a busy world....The death of the mortal body does not bring a death of the spirit.