> When archaeological evidence shows that people largely stopped farming, it usually means that those people largely stopped existing

No, there are signs of habitation. If you look at the evidence from the alluvial plain around ancient Babylon and similar cities from the early history of cities in the region you see people returning to fishing, hunting, and sort of opportunistic planting on river banks. These cities had a tenuous grasp, and people would often just leave, or the city would collapse and people would stop doing intensive agriculture. People would abandon irrigation channels which they had been forced to dig and simply just go fishing. James C. Scott's Against the Grain covers the archeology nicely.