Why does the Old Testament never mention unclean spirits and demon-possessed people? Or I am missing something?
You're missing something. :)
1 Samual 16:14:
Then he started playing a game of darts with David as the bulls-eye.
As the light shines more brightly the shadows appear more pronounced.
With the Christ event the Kingdom of God has come. It is evidenced by Him casting out demons by the finger (=authority)/Spirit of God. Thus demons are exposed when they come close to Christ and the Spirit filled community.
I would suggest reading some work by George E Ladd on the Kingdom of God in order to get a more thorough understanding.
Another aspect is that the Old Testament texts do not have as a purpose to teach demonology. There are lots of stuff they are leaving out since it is of no value to the point the texts are trying to make. In the original context the authors wanted to teach the listeners (few people would read the texts before Gutenberg) about faithfulness to the Covenant between God and his people.
And yet another aspect is that language itself had to evolve. The prophets spoke words that made sense to their original audience. That audience had no understanding of the Bible in its complete form. It was not until the inter-testamental period that there was a nomenclature available to speak about demons.
So instead of using technical terms, they are using symbolic language: "great waters" and other various descriptions of powers of chaos.
This also helps explain why Saul could be tormented by a spirit "from the Lord". The purpose of that text is not to teach the chain of command in the spiritual realm or the mechanism of abandonment. Fixating upon grammar, especially in a translation that can not do justice to the Hebrew text, is a sure way to miss the point of the text. "from the Lord" is not a terminus technicus.
I wonder if the "deceiving spirit" of 1 Kings 22:22 would count? There's also an "evil spirit" in Judges 9:23, and then there's Saul's "evil spirit", as David Morton mentioned.
Satan is mentioned in a number of places (such as 1 Chronicles 21:1) and some attribute his name to his minions as well, so such references could be to demons as well.