Is the TOE falsifiable and if it was, would it advance Biblical Creationism

What I find puzzling, is when creationists assert both

The theory of evolution is false; and

The theory of evolution is not falsifiable.

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What can they be thinking?

That would depend on how ToE were falsified. A clear proof of creationism might falsify ToE, and would surely provide credence to creationism. However, such a proof seems unlikely.If ToE were to be falsified, it is most likely that what would be shown is evidence that is contrary to some technical aspect of the processes assumed to be responsible for evolution. That kind of evidence would not prove creationism, and might merely require that ToE be modified to fit the new evidence.

That's a misunderstanding of "unfalsifiable". The point of falsifiability is to rule out statements such as:

All bachelors are unmarried men.

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That is, the idea is to rule out statements which are necessarily true by virtue of the meanings of the various words used. An unfalsifiable statement would be necessarily true, and no evidence could ever show otherwise.

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It makes no sense to say that ToE is both false and unfalsifiable.

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It's unfalsifiable because it's mostly hypothetical scenarios treated as fact.

That would make it true but of no scientific relevance. However, that's a gross mischaracterization of ToE.

There are people who believe the book of Mormon. And I presume that believing the book of Mormon is knowing it to be true. So that makes the Book of Mormon really great evidence for the past. I guess the Koran is also really great evidence for the past. Maybe Grimm's Fairy Tales is also great evidence for the past.