As I understand it, there are a few reasons the Genesis creation story cannot coexist with evolution. One is that Adam's transgression brought death into the world, but evolution requires a long string of successive generations, which means death. Another is that Adam is created as the first man, but evolution would have produced a long string of humanoid creatures that has led to our modern selves.
Here's a theory that I think satisfies the requirements of both Genesis and evolution.
Life evolved over a long period of time, leading to a number of humanoid creatures. At some point, one is sufficiently advanced for God to say, "I can interact with this being as a potential heir." This being is Adam. He is the first entity to get a human spirit. Thus Adam's body was the result of evolutionary processes, but God created Adam as the first man.
As the first being with a human spirit, Adam is given commandments that the rest of creation had not received, as they were merely animals. Through his transgression, Adam brought spiritual death--the only type of death that matters to God--into the world. (See Doctrine and Covenants 29:41-43, where God says Adam's transgression made him spiritually dead, and Doctrine and Covenants 101:29-30, where God says "there is no death" in the Millennium and then talks about "an infant shall not die until he is old"--so people are dying during the Millennium but this isn't what God considers "death.") Thus evolution had its long string of successive generations but Adam brought death into the world.
Adam's posterity was commanded to not intermarry with the humanoid creatures around them, and Satan tempted the sons of Adam to ignore this commandment, which gives us the formulation from Genesis 6:2 and the Book of Enoch about "the sons of God" and "the daughters of men." Those who didn't intermarry were given to the alternate extreme, extermination. This fits with the archaeological record which shows Homo sapiens killing and interbreeding with other humanoids, most notably Homo neanderthalensis.
I think this way of looking at things allows for the aspects of human history that have the most-compelling scientific evidence while not doing such violence to the concepts of "creation" and "death" that the Bible is rendered meaningless.