Shakespheare wrote plays...he never claimed they were true.
To the contrary, a number of his plays portray English historical events. They're not presented with disclaimers that deny their historical accuracy. Neither, of course, does he claim it literally happened that way.
Is Shakespeare a liar?
The Bible claims to be "Truth".
No, the Bible claims to be God-breathed, and profitable for reproof, doctrine, and teaching, so that the readers of the Bible may be righteous.
It does not claim to be a history textbook. That's your erroneous interpretation. It does not claim that the account of Genesis is literally how it happened.
Archeology has proven many of the historical facts and has never proved it wrong.
Archeology proves that the Flood never happened. That's one thing, right there.