I see in scripture (see citations below) some instances of evil spirits. How did the priests (I'm guessing it was the priests as they also dealt with instances of leprosy) deal with people that were possessed or tormented by evil spirits?
You cite a number of verses, but none of them touches on your question ("How did the priests… deal with people that were possessed or tormented by evil spirits?") except Ⅰ Samuel 16:14–23. There, the "evil spirit" is exorcised (if you will) not by any arcane ceremony, and not by a kohen ("priest"), but by the playing of music. (See verses 16, 23.) Seemingly, the "evil spirit" was a form of depression. This is, indeed, implied by the M'tzudas David to verse 16, who writes that "because of the joy of the sound of the music, the dread will pass". (That it was "from God" (verse 17) means just that — that God instilled it within Saul — but nonetheless it seems to have been some sort of depression, and nothing that begs arcane explanation.)
I found a relevant section in the Jewish Encyclopedia:
Josephus ("Ant." viii. 2, § 5) relates:
"I have seen a certain man of my own country, whose name was Eleazar, releasing people that were demoniacal, in the presence of Vespasian and his sons and his captains and the whole multitude of his soldiers. The manner of the cure was this: He put a ring that had a root of one of those sorts mentioned by Solomon to the nostrils of the demoniac, after which he drew out the demon through his nostrils; and when the man fell down, immediately he abjured him to return into him no more, still making mention of Solomon, and reciting the incantations which he composed. And when Eleazar would persuade and demonstrate to the spectators that he had such a power, he set a little way off a cup or basin full of water, and commanded the demon, as he went out of the man, to overturn it, and thereby let the spectators know that he had left the man; and when this was done the skill and wisdom of Solomon were shown very manifestly." See Ba'aras.
Exorcism in Rabbinical Literature. Rabbi Johanan ben Zakkai, a contemporary of Josephus, alludes to the practise of exorcism by saying: "Has an evil spirit never entered into you? Have you never seen a person into whom an evil spirit had entered? What should be done with one so affected? Take roots of herbs, burn them under him, and surround him with water, whereupon the spirit will flee" (Pesik., ed. Buber, 40a). R. Akiba (d. 132), in speaking of diseases, uses the technical terms of exorcism ('Ab. Zarah 55b). Simon ben Yoḥai drove out the demon Ben Temalion from the daughter of a Roman emperor (Me'i. 17b).
The verses cited do not discuss demonic possession. Those discussing demons are condemning people for worshiping false gods.
The verse cited that discusses "priests" and "possession" and "casting off" is discussing the Priests and Levites who were cast off by the king from the land they possessed.
The verse that talks about Saul, which you have interpreted to be discussing his being terrified by an evil spirit, is talking about his being filled with dread because he had lost G-d's support and he knew it. His advisers told him that he was possessed by an evil spirit sent from G-d, but he knew in his heart that his reign was over, and he was terrified of what was to come (and, it turns out, rightly so). He asked for someone to play music to calm him down. David was known in the camp for his talent, so they chose him to play music. He was not a priest, and he was not chosen because he could perform an exorcism. There was no ritual being performed here.
In short, I do not believe the Priests (Kohanim) ever performed a ritual exorcism of demons possessing a human. In this answer I am attempting to point out that the idea is not borne out of the verses cited as the basis for the question. I hope this helps to clear up any confusion in the verses.