I'm wanting to know about any possible demonic oppression and/or possession in the Old Testament, as well as why demons being cast out is not done before the time of Christ?
Was it because Christ as Savior had not appeared yet?
I am going to recommend you watch Dr. Michael Heiser video, Demonic Entities & The Messianic Profile | LXX, DSS, Psalm 91 & 151.
1) King David is credited for writing 4 Psalms to be "performed over the possessed" (in Hebrew the word "assaulted" is used) according to a passage of a Qumran Dead Sea Scroll.
2) King Solomon in Jewish mystical tradition is also accredited with having knowledge of exorcism and there also is Qumran, Dead Sea scroll passage that speaks about that and Josephus also confirms that in his Antiquities.
3) Heiser does acknowledge that the prolific exorcism activity that we see Jesus doing would have been a sign of him being the messiah. Psalm 91 especially has connotations for exorcism, especially in the Septuagint translation. "Terror by Night" is a cultural reference to demons that afflict people at night and "the arrow that flies by day" is rendered "noonday demon" in the Septuagint. Anyway the terms used in Psalm 91 especially makes authority over demons part of "messianic" profile as Heiser describes it (because only the Messiah would have the authority over these beings and in the near east some of these beings were actually considered to be deities by the surrounding pagan peoples.)
Personally, I believe the difference in the Old Testament and New Testament on this subject comes from the difference in covenant and dispensation that we see in other topics like salvation. God also used the demonic activity of the New Testament to be "a foil" to highlight Christ's unique identity and ministry.
Demon Possession in the Old Testament?
We can be sure that demonic possession occurred prior to the birth of Christ. There are a few instances in the Old Testament of demonic activity in the affairs of people.
Exorcism is a ritual of power performed in order to drive an evil spirit, whether demonic or ghostly, from a possessed person, location, or object. The Christian scholar Origen credits Jews with a special talent for exorcising demons (Against Celsus, book 4).
The first allusion to exorcism appears in the Bible, in the youth narratives of David (l Samuel). But while the biblical David seemed to be able to effect a temporary expulsion of Saul’s evil spirit using music, the book of Tobit contains the first explicit description of an (informal) exorcism. Josephus recounts incidents of possession and exorcism in his Antiquities of the Jews (2, 5, 8, 45-48). In his description, exorcism involved burning herbs and immersing the possessed person in water. The New Testament also reports Jesus to have performed numerous exorcisms of demonic spirits in first-century Palestine (Matthew 12; Mark 5, 6, 13; Luke 8).
The Dead Sea Scrolls include several exorcism incantations and formulae, mostly directed against disease-causing demons. The DSS Psalms collection in particular (11Q5) has “four songs for the charming of demons with music.” People who fell under the influence of false prophets and mediums were thought to also require the exorcism of possessing evil spirits (the false prophets and mediums themselves were subject to death, a sure cure for most possessions; see Zechariah 13). - Jewish Exorcism
There is at least one New Testament source admits that the Jews definitely performed exorcisms:
When Jesus sends out the Twelve Apostles, “they went out and preached that men should repent. And they cast out many demons, and anointed with oil many that were sick and healed them” (Mark 6:12-13). But did you know that exorcisms actually predate Christianity, and that there were Jewish exorcists at the time of Christ?
Jesus actually refers to this, but it’s such a brief mention in Scripture that it’s easy to overlook. It’s in Matthew 12:24-28, while the Pharisees are questioning Him about His own exorcisms:
But when the Pharisees heard it they said, “It is only by Be-elzebul, the prince of demons, that this man casts out demons.” Knowing their thoughts, he said to them, “Every kingdom divided against itself is laid waste, and no city or house divided against itself will stand; and if Satan casts out Satan, he is divided against himself; how then will his kingdom stand? And if I cast out demons by Be-elzebul, by whom do your sons cast them out? Therefore they shall be your judges. But if it is by the Spirit of God that I cast out demons, then the kingdom of God has come upon you.
Jesus’ remark makes it clear both that Jewish exorcists existed in His day, and that they were apparently successful in driving out demons.
That’s all the detail that Scripture gives us. Fortunately, the first-century Jewish historian Josephus tells us much more, in Book VIII of Antiquities of the Jews. - Casting Out Demons in the Name of Solomon: Jewish Exorcisms at the Time of Christ
Demonic activity in the Old Testament:
There seems to be at least vague two incidents in Scriptures for those who are of Catholic and Orthodox traditions.
The first one can be found in 1 Sam 16-23: The first incident involves David where he plays the lyre/harp (1 Sam 16: 18-23) in Saul's presence and the Spirit of Melancholia could flee his presence.
Whenever the spirit from God came on Saul, David would take up his lyre and play. Then relief would come to Saul; he would feel better, and the evil spirit would leave him (1 Samuel 16: 23).
Fr. Gabriele Amorth SSP mentions in this in one of his books that the singing of sacred hymns is a valid form of expelling the demon. Being a Catholic priest moreover, he encouraged the singing of Gregorian Chant during exorcisms, since Gregorian Chant in itself is very scripture based. Fr. Amorth makes note of David playing the harp incident.
When they had finished eating, they escorted Tobi′as in to her. As he went he remembered the words of Raphael, and he took the live ashes of incense and put the heart and liver of the fish upon them and made a smoke. And when the demon smelled the odor he fled to the remotest parts of Egypt, and the angel bound him (Tobit 8).
A second incident comes from the Book of Tobit which is part of both the catholic and Orthodox biblical canons. The Book of Tobit recounts the story how the young Tobias with the aide of the angel Raphael made the demon flee their presence.
Fr Gabriele Amorthe mentions in one of his two books (on exorcisms) this incident and encourages the use of blessed incense in homes subject to demonic influence.
Josephus Flavius ("Ant." viii. 2, § 5) relates the following:
"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. - Jewish Encyclopedia