Sumerian culture was many thousands of years after the end of the ice age. Even the last big cooling event was a couple thousand years before Mesopotamian cities, and that was just a cooling event, not something with big ice sheets that turn into floods when it warms up. What they had is the Tigris and Euphrates that did flood on a regular basis, sometimes catastrophically - the floods of the Nile brought fertilization from the upper terrains they covered and it was predictable like clockwork, but the floods of Mesopotamia were destructive and unpredictable. One thing it absolutely didn't do is cover the whole Mesopotamian plain, it just flooded the land surrounding the river. Unfortunately, people make cities near those rivers - but the mountains were WAY too far to run to them. Mesopotamia is just basically one gigantic flat plain, it doesn't have random mountains in the middle.

We have geological records of one big flood dated around 2900 BCE that destroyed most notably Shuruppak (it got better), which held a big cultural position at the time, and a few other cities in the area. What's funny is that by the time the Flood story was integrated into Akkadian / Babylonian culture, sometimes between 2000 and 1800 BCE, there were still people living in Shuruppak, which is named in those myths as having been destroyed.