What I’ve noticed, however, is that philosophers tend to make a few mistakes when it comes to the relationship between DCT and moral arguments for God (MAFG).
One mistake is to assume that if DCT is false, then all MAFG fail. However, this is not obviously true. It’s true that a lot of moral arguments would go down the drain, but there are some moral arguments that would seem to still remain. For example, it seems that arguments from moral knowledge/awareness for God’s existence would still remain. That doesn’t mean that said arguments are good arguments, rather, it just means that certain objections (like the Euthyphro objection) don’t apply.
Another mistake is to assume that it’s obvious that the Euthyphro objection (EO) to DCT is a bad objection. It’s not obvious (to me). It’s arguable that the typical theist response (see below) just reframes the problem.
The third mistake is to assume that if one has dealt with the EO, then DCT is saved. This doesn’t follow. In fact, there are other objections to DCT. In fact, I don’t even think an objection that was made by J.L. Mackie (in the 1980’s) has ever been satisfactorily answered. Mackie’s objection is something like, “Why ought we follow God’s commands? What obligates us to follow what God says?”
The Euthyphro Dance 
Theists typically says God is The Good; God’s nature is good. Even if we accept that God is all-good by definition, it doesn’t follow that this means God grounds morality by definition (that would beg the question). It’s hard to see how that would work because if moral realism is true, moral truths, at bottom, are necessary truths. Given that they are necessary, it’s far from clear why we would even need God to account for moral facts. Not to mention, there are atheistic accounts of moral facts.
If God is The Good, then one is saying that if God exists, then God is The Good. The problem here is that (most) moral arguments are now rendered completely circular, because the moral argument is trying to show the God exists!
In response, theists like William Lane Craig typically say that God is a necessary being. Thus, God exists in all possible worlds. But even if God exists in all possible worlds, this doesn’t explain why God is needed. Even if God could ground morality, it doesn’t mean that God is required. Not to mention, there are logically possible worlds where God doesn’t exist, because God does not exist by definition; the view that God exists by definition is implausible, and that would render the moral argument irrelevant.
In addition, some theists (like Richard Swinburne) conceive of God as a contingent entity.
Speaking of Craig, it appears that one doesn’t even need to use the EO (or object to DCT) in response to Craig’s argument. That’s because Craig’s moral argument is extremely counter-intuitive. Craig himself admits that his argument entails that moral facts can’t possibly exist without God existing. One problem is that this doesn’t seem logically impossible. It doesn’t seem impossible that murder would still be wrong if God doesn’t exist. Secondly, as mentioned before, there are non-theistic accounts of morality. Are these accounts somehow (all) logically incoherent? Of course not! Thus, a more modest moral argument would say that God is the best explanation of moral facts .
 Coined by “Counter-Apologist”
This post isn’t dedicated to the subject of whether or not moral facts exist (i.e. moral realism). I myself am skeptical about whether there really are moral facts. I don’t claim that there aren’t moral facts, instead, I am on the fence. Even if one thinks that some moral propositions are “true”, that doesn’t mean one is committed to moral truths actually existing “out there” in the universe.