I am not sure of dates but it seems that the church existed before slavery was fully abolished. Are there any instances known about where members had slaves?
Yes, the Church was established during the antebellum years (the years before the Civil War) and both Joseph Smith and Brigham Young lived during that period. (Joseph Smith prophesied of the coming Civil War. Doctrine and Covenants 87).
While the Church was established in New York and Illinois, there could not have been slaves, because these states were anti-slavery. Thus many of the first members of the Church were from the North and tended to be anti-slavery. When the body of the Church moved to Missouri where the persecution they were suffering increased, it was in part because the pro-slavery Missiourians feared that the mostly anti-slavery Mormons would swing the vote.
When Joseph Smith ran for President of the United States, part of his platform was a plan to eliminate slavery. He suggested we “pay every man a reasonable price for his slaves out of the surplus revenue arising from the sale of public lands, and from the deduction of pay from members of Congress.” Joseph Smith: Campaign for President of the United States
Slavery was legal in Utah, due to the Compromise of 1850. When the Saints moved to the Utah territory, there were three slaves, Green Flake, Hark Lay and Oscar Crosby in the first pioneer company in 1847. When slave owners converted to the church, some gave the slaves their freedom before migrating, or after arriving in Utah, others brought their slaves with them
One family freed their slaves, but because the woman was a widow, one of the slaves, “Faithful John” traveled across the plains with them to help the widow and her children make the trip. Once they were arrived and established, he set out to return to the East to look for his wife (who was sold by his previous owner.) However, the family later received word that he had died before reaching the East.
One family brought their slave, a woman named Biddy Mason. She was born in Missouri and “belonged” to Robert Marion Smith. When Smith and his family joined the church they took Mason with them to Utah. She herded cattle, cooked, served as a midwife and tended her own children. In 1851, Robert Smith got gold fever and went to California. Apparently, he was unaware that California was a free state. In 1856 Mason petitioned the court and was able to obtain freedom for herself and her daughters. She went on to become a successful entrepreneur.
There likely were other slaves in Utah as well, but I believe this was the exception, and not the rule. It is notable that there were free black people who joined the Church as well. Elijah Abel, the first black man to receive the Priesthood (from Joseph Smith) and Jane Manning James are two of our black pioneers.