I hardly remember, but if I do recall correctly... In the beginning of the story a man, a soldier, is in a war and enters some sort of time warp by mistake. He ends up in some sort of fantasy world being chased by a Dragon breathing fire instead of a missile and is saved by a dwarf man or an old man or something like that. Then the story switches to another character and talks about him being a dark elf and he lives in a village were everyone hates him. The king holds a competition for the villagers to win his daughters hand in marriage and the Drow fancies her quite a bit and enters. After slaying the knight he finds that she is the knight and she says that she would rather die than be with him. And he becomes evil. Mean while the man who came from our world is slowly becoming a white elf or is a white elf or something like that. If you can help me find this book I would greatly appreciate it.
The Bifrost Guardians - Book 1
Torn away from the midst of a firefight in Vietnam and catapulted to a time and place where the Norse gods fight a deadly war of their own, Al Larson finds himself locked into an elvish body on a world where swords and spells are the means of battle. Al must adapt swiftly - or die. For the gods have marked him as their own private battleground, and Al's only chance rests in completing the quest Freyr has set him, a quest that will lead him to the very gates of Hel, where he must save a god - or destroy one! See the Author's list of her Novels.
The character starts out in Vietnam and is magically transported due to praying to a Norse god into the body of an Elf. He is helped by a magician and her Samurai helper. They battle her brother a dark elf who tried to marry a princess but was shunned.
I think this is your book. It matches enough of what I remember from reading it to match your description but the story as you tell it is a little different.