Understanding the distinction between contra-inerrancy arguments, and arguments that lead to a positive conclusion (in the context of the Gospels) was quite possibly the final “aha” moment for me on the way out of Christianity. With the second approach, to goal is not to find contradictions but to find clues that help us figure out the history of the writing of the Bible. Once dozens of these clues are harnessed together by all supporting the same point, they cannot be belittled one piece at a time as trivial details or something for which an explanation will present itself at a later time.
I would like to begin my argument by pointing out that the prima facie case is mine. Read Genesis 1:1-2:3 by itself and you have a complete story of the creation of the world and everything in it. Read Genesis 2:4-2:25 and you have a complete story of the creation of the world and everything in it. Both stories are begun with verses that would make perfect sense as the first verse in a book.
If these really are different stories, what we should expect to find is differences in the details that must be explained away to maintain that it's really all the same story. If this really is the same story, we should expect the halves of the story to refer to each other in ways that just don't make sense when viewing the stories as individuals – especially because the second half is claimed to cover a time interval contained within the first half. These are the criteria by which I will be making the case for two different stories.
(Don't think that I think I'm some scholar who knows the official criteria by which this is normally judged. I'm merely spelling out precisely what I consider to be common sense so that if anyone disagrees with my overall approach, it's clear what they are arguing against.)
Man Before Plants?
Plants are created on the third day, which is certainly before the creation of people on the sixth day.
However, Genesis 2:5 lets us know that plants are not created yet. “Now no shrub of the field was yet in the earth, and no plant of the field had yet sprouted, for the LORD God had not sent rain upon the earth, and there was no man to cultivate the ground.” Before plants are created in 2:8-9, Adam is created in 2:7. The prima facie case is mine again: Adam is created before plants in the second creation account.
Looking closer at the logic of the story makes the point even more clear. In 2:5, we are given two reasons for a lack of plants: no water, and no man. 2:6 solves the first problem as a mist comes. 2:7 solves the second problem as man is created. Both problem are now solved. So God is now ready to plant a garden and make plants grow, which he does in 2:8-9. It's not just the order in which event are recorded that suggest Adam was created before plants, but the logical flow of the account as well.
Creationists' rebuttal is that Genesis 2:5 refers only to specific kinds of plants, namely cultivated plants. Thus, most of the plants were created on the third day, while the cultivated plants of Eden were created after Adam on the sixth day. I'm no Hebrew scholar, but just looking up all the different words used for shrub and plant in Strong's Concordance offers absolutely no support for this position. I see no reason to think Genesis 1:11-12 excludes some kinds of plants and I see no reason to think Genesis 2:5 includes only the kinds of plants not created on the third day. Without either of these arguments, the YEC position fails without even looking outside the Bible.
The only reason I see for even speculating about either is simply that it is needed to make Genesis 1-2 flow as a single story. Another way of saying this is that the creationist position is to begin with a certain conclusion and then look for an interpretation of the words to make it work. But that's not how you're supposed to read things when the goal is a truth search and not merely the affirmation of preconceived ideas. The intellectually honest approach is to let Genesis tell you what Genesis is saying. The way young-earth creationists cannot do this is precisely the kind of bending over backward that should be expected if there really are two creation stories.
Also, this doesn't make sense of the logic of 2:5-9. The reason for no plants of some kind is a lack of rain and a lack of man. Now, what kind of plants either need rain or need man? Pretty much all of them, at least according to non-technical ideas of what a plant is. If there's a distinction between wild and cultivated plants, then I would guess that no rain is why there are no wild plants and no man is why there are no cultivated plants in 2:5.
Animals Before Eve?
On the sixth day, God first creates the animals, and then he creates people. However, in the second creation account, the order is Adam, animals, Eve. Creationists generally agree with the first part, so I won't belabor that point.
The order in which the events are recorded is the creation of Adam (2:7), animals (2:19), and finally Eve (2:22), so the prima facie case is mine again. But the case is much stronger than the mere order in which the facts are recorded – this is the order that is implied by the logic of 2:18-2:22. In 2:18a, God observes a problem: man is alone. In 2:18b, God suggests a solution: Adam needs a helper. The next thing that happens is God creates the animals in 2:19 as an attempt to find Adam a helper. My claim that the creation of the animals was an attempt to find a helper for Adam is all but explicitly stated in 2:20: “but for Adam there was not found a helper suitable for him.” In verse 2:21, God tries a more successful solution: taking one of Adam's ribs and making a woman.
To argue that Genesis 1-2 is a single literal account means that three things that must be explained away. First, the order in which the events are recorded must be overlooked. Second, the awkward insertion of the story of the creation of the animals (2:19-20) into the story of the creation of Adam's helper (2:18 & 2:21-22) must be ignored. And finally, Genesis' own explanation for why the creation of animals fits into the creation of Adam's helper must be ignored. I don't see how this position can be held unless one is taking the approach that Genesis must be true and literal therefore there must be some way of resolving the contradiction.
This is precisely what should be expected if these are two different creation accounts that were not written to go together. The six days of creation have their own logical structure, and the second creation account has a logical account of needs and solutions. Each makes sense alone, but to view them as going together prevents the reader from seeing what the second author is saying.
Two Creation Stories/Two Flood Stories/Two Authors
The case for two authors gets even better when compared with the two flood stories. In one of the flood stories, God was known as Elohim, while in the other, he was known as Yahweh. In all 38 instances, the God of 1:1-2:3 is Elohim. In all 11 instances, the God of 2:4-2:25 is Yahweh.
Also, in the flood stories, it was the Elohim author that spoke of the opening of the windows of heaven and fountains of the deep, while the Yahweh author says the flood comes because it starts raining. One is giving more of God's perspective while the other is giving more of man's perspective. We see the same thing with the creation accounts. The Elohim author doesn't even mention people until the end, and then man fades into the background again as Elohim rests. The Yahweh author describes the creation of plants and animals in the contexts of plants needing man and man needing a helper.
Furthermore, on the second day of creation, Elohim creates an expanse and calls it heaven. Water is created that is above this expanse. It was Elohim who opens the windows of heaven to let this water out to flood the earth. This suggests that not only are there two authors of the creation and flood stories, but they are in fact the same two authors.
If Genesis 1-2 is a single narrative, it is quite curious that we find internal references and similarities between certain halves of the creation and flood accounts, but we don't find internal references between the two halves of the creation account, in spite of the fact that the time interval of the second half is completely inside the time interval of the first half.
I encourage you to read creationists' rebuttal to these arguments, because it so clearly shows that the arguments I am presenting have been noted and answered poorly. Both sides have reasonable positions if you are just reading an overview of the positions. Where creationists lose is when you compare each position to the specifics of the details in how Genesis 1-2 is written.