On reading your post I’m unsure of what you are arguing here. I’ll put two possibilities to you and address them:-
I'll pick up on the main points of your post. Option #1 here is not my position.

Option 2. God is logic and because God is unchanging then logic is unchanging.
This appears to be how you are arguing, although I’m a bit unsure. If so, then it’s an assertion you make, or if not one you make because you read it from the Bible, then it’s an assertion someone else made, an assertion which you accept.
Under that circumstance, I fail to see why I also cannot argue by assertion. That is, I make my assertions, stand by them, and say that they account for logic.
For me, the universe could not make sense if things could both exist and not exist, and it seems to be that way for you, and for God.
And why exactly should logic change tomorrow in my universe? And why should God not go back on his word in your universe?
If God cannot go back on his word because the universe would not make sense, then you seem to be saying that God himself is somehow constrained, and this brings us back to the point 1) I made above. Logic appears to be independent of God.
If you insist that God is not constrained, then it seems to violate the principle which I think you could be arguing here - God is logic and God/logic cannot change because if they did, then the universe could not make sense.
I agree. Logic cannot change otherwise things don’t make sense. But I do so with out asserting the existence of God. I assert that the universe exists and that logic is a part of it.
IOW I continue to think that when it comes to accounting for logic, even from the perspective of a logical argument, we remain in the same boat.
Let me see if I can also address this in a flowing manner so as not to have it broken up and blowing out! First, the nature of God. Yes, there is a “constraint” on God. He is who He is, and He cannot be anything other than Himself. He cannot lie. He cannot contradict Himself. He cannot change His nature. Part of His nature is logic. He is not independent of logic; logic is how God thinks, and because He cannot change then logic (His way of thinking) cannot change. So, because God exists and is perfectly logical and made us in His image, we are to think His thoughts after Him.

Yes, the Bible asserts that God's thoughts and ways are perfectly, consistently logical, good, and right. It doesn't defend the position because no defense is needed. The Words are God-breathed, ultimately it's His message and not just messages from humans. The Bible does not seek to defend what it says – it asserts the truth of itself being from God.
You believe what other people tell you (or you believe what you deem right). We are all fallible humans who make mistakes and change our minds about what is right or wrong, good or bad, etc. God does not do that. He chose to reveal that to us in a book. We can believe what He says, or we can believe what humans say. When we believe God and pattern our thoughts after Him, we can be consistent with our logic and lifestyle. A person who refuses to believe God's Word patterns his thoughts after himself or other people. What person is perfectly logical, that we should pattern our thoughts after him or her?
You and I agree that logic is universal and will not change. However, let's step into a godless universe and see if that assertion could hold true.
(1) “People have made up logic.” If so, then they can change it. No one person has the authority to define logic for every other person. Plus, as we evolve into higher intelligence we can change logic to make it fit with the intelligence we think we are gaining. Also, there is no reason logic would be universal if we just made it up. Different people groups with different ideas on thought processes could develop their own logic. Talk about a mess when communicating – each person could use his own arbitrary logic!
(2) “We just evolved this way, to think this way, which is universal logic.” OK, that's what we observe, but that doesn't account for (justify/give a good reason) why logic exists in the first place. And if we just evolved this way, which would be chemicals reacting, we could only think according to how the chemicals fire and have no independent thought processes. You believe there is no god because that is how your chemicals fire, and I believe God exists because that is how my chemicals fire. Therefore, neither one of us would have a reason to argue our position (except because our chemicals tell us to) and post-modernism is true....which is self-refuting, illogical.
(3) “The universe exists and logic is part of it. God need not exist for logic to exist.” Arbitrary assertion. I gave a good reason as to why the God of the Bible must exist for logic to make sense, therefore being non-arbitrary. And I showed in points 1 and 2 above why the typical atheist reasons for “logic but no God” don't work – they defy the logic the atheist asserts and assumes at the outset! One more........
(4) “Neither one of us can adequately explain the existence of logic.” That's the Tu Quoque fallacy. Even if you don't agree with my reasoning (which I don't expect you to unless you start with God's Word instead of human ideas), I still provided a logical reason for laws of logic to exist; and I showed why the evolutionary ideas don't work, as they end up being illogical when thought out to their logical conclusion.

... can you give me a good reason for why logic even exists in the first place if the ToE were true?
Well yes. I can get part of the way there at least.
If an organism cannot make rudimentary logical judgements at least, either by intelligent judgement or hard wiring then it ceases to exist. Either food is there or it is not. Either a predator is there or it is not. Either a mate is there or it is not. If a predator is there, then run in opposite direction. If food is there, then move towards it.
The differences here are: we don't assume food to eat it; we don't assume a predator for it to be there; we don't assume a mate to have one; we HAVE to assume logic in order to use it. Also - food, predators, and mates are physical in nature, whereas logic is conceptual. I'd say this is a category mistake, to compare logic to physical things. Care to try again to give a good reason why logic exists if the ToE were true?

But be careful. Here you appear to want me to give you a mechanistic description to account for the existence of logic. If so, then I will need to ask you for one.
No mechanistic descriptions - logic is not physical in nature! But if you think it's physical in nature, I'd love an explanation!