—Thought is merely the byproduct of some atoms within my skull.—
There is no reason to conclude that that is the only possible explanation, SIMPLY because you posit that no one designed your brain (indeed, even evolution suggests that your brain is the result of a specific and historical process, not simply a random collection). For all anyone knows, it could be the byproduct of etherions moving about in a mystical realm.
But if it isn’t a product of SOME process, then what is it? Simply saying that God designed your brain doesn’t in the least explain what thought IS, or even help you determine whether thoughts are just the byproducts of atoms in your skull. Your brain could just as easily have been designed to think BY a God that used atoms in a particular way to produce thought processes.
(by the way, would you think that “a hunk of metal” is a helpful way to describe a computer when talking about what it can do?)
—But if so, how can I trust my own thinking to be true? But if I can’t trust my own thinking, of course, I can’t trust the arguments leading to atheism, and therefore have no reason to be an atheist, or anything else.—
No one is trying to make you be an atheist. Even a totally undirected process of evolution could be equally consistent with a god designing everything: because presumably a god would know the exact outcome of any sort of set of initial conditions it created.
Regardless, you cannot make a conclusion like “my brain was designed for thinking” just because you don’t happen to LIKE some of the potential implications. What does that have to do with anything? I don’t like it that a mass murderer rules Iraq… but that doesn’t change the fact that one does.
—But if so, how can I trust my own thinking to be true?—
What do you mean by “true” in this context? Do you mean that the arguments you might make can validly lead to true conclusions (I would argue that they would be regardless of anyone “thinking” them or not, since truth is an operational definition)? Or do you mean that your thinking is “real”?
How can you trust that your own thinking is “true” (whatever you mean by that) right now?