S5 of modal logic states, essentially, that if possibly necessarily A, then necessarily A.

Okay, I think the crux of your issue is a slight misunderstanding of possible world semantics. A necessary truth, as you say, is one that is true in all possible worlds. From this it immediately follows that if we find a possible world wherein there is a necessary truth, that this necessary truth is in all possible worlds. I hope this makes sense: to make it even clearer, let us consider an example. Suppose we have three possible worlds W1, W2, and W3, and we have a necessary truth T1. By definition, T1 is in W1, W2, and W3. Now, suppose we are simply looking at W1. We see that T1 is in it and that it is a necessary truth. So, we deduce that T1 is in W2 and W3 as well. I hope this provides some clarity, and shows how straightforward the claim S5 is making is.

I don't see how we could ever know that a thing is "possibly necessary" without first knowing that it is "necessary."

This need not be a concern, because being possibly necessary (as you say) and being necessary are one and the same.

Otherwise, couldn't anything be "possibly necessary," and thus every conceivable thing necessarily exists.

No, it couldn’t. As before, if statement is possibly necessarily true means that it is true in every possible world, which is to say it is not conceivable for it to be false. And very few propositions satisfy that test.

Just to sum everything up, I think your confusion stems from the use of the term "possibly necessary." This does not mean that it is possible that it is true in every possible world -- that is confused. If it is true in every possible world, then there doesn't exist a possible world where it is false. But this contradicts the fact that it is "possibly." (This is what I think you are getting at in your argument). Rather, "possibly necessary" simply means that it is true in every possible world, or in other words simply necessary.

I hope this clarifies matter! Let me know if you have any questions/concerns.