Poor old Noah spent all those years building the ark when all he had to do was pick up and move a few miles. :^)
If Noah's flood was a localized, there would've been no need to build a ship. The Black Sea may have flooded, but it wasn't Noah's flood.
At the end of the Ice Age, you're correct, everything was flooding. The Persian Gulf was completely dry, the Red Sea was land Locked as were parts of the Mediterranean (divided in sections), flooding each one after the other and then finally the Black Sea. There would have been flood refugees everywhere, I mean, everywhere. When Noah heard all the flood stories, he began to build an ark, a big one that took a lot of time...and likely up on a mountain, it may still be there.