Delaware had already effectively abolished slavery decades before the Civil War; while still technically legal, the sale, purchase, or importation of slaves was banned and by the beginning of the war only a few dozen "indentured apprenticeses for life", the state's legal term for slaves, still existed. It should be noted however that Delaware authorities had a reputation for harshly enforcing runaway slave laws; that the state held a secession vote (it failed over the governor's objections); that a not insignificant number of Delaware residents fought for the Confederacy in the war; and that Delaware voted against ratification of the 13th Amendment, which banned slavery.
Kentucky declared official neutrality at the outset of the Civil War, but after a failed attempt by Confederate General Leonidas Polk to take the state of Kentucky for the Confederacy, the legislature petitioned the Union Army for assistance and after early 1862 the state was effectively under Union control. In Kentucky, more than any other state, the scenario of brother against brother played out, with large numbers of men fighting on both sides. Kentucky citizens also formed a Confederate "shadow government" which existed throughout the war, and the state was actually admitted to the Confederacy in late 1861, becoming the center star in the battle flag.
President James Buchannon, Lincoln's predecessor, understood the very real threat of Maryland's secession and filled the state with federal troops. Maryland was economically and socially divided, but was overall more opposed to abolition than for; Lincoln won only 2.5% of the vote in the state in 1860 and approximately a quarter of all Marylanders who fought in the Civil War did so for the Confederacy. I will also note that the state song "Maryland My Maryland", written during the runup to the war, includes lyrics urging Marylanders to "spurn the Northern scum" and "burst the tyrant's chain" - in other words, to secede from the Union.
Virginia did secede secede, becoming the 8th state to join the Confederacy, and for most of the war the Confederate capital was in Richmond, from May of 1861 until just before the city's fall in April of 1865. Perhaps you were thinking about the northwestern counties that themselves seceded from the state to form West Virginia?
Missouri sent armies, generals, and supplies to both sides, was represented with a star on both flags, maintained dual governments, and endured a bloody neighbor-against-neighbor