They did not truly use the same script the entire time. Like all writing systems the script did change as people simplified characters for their own convenienc

They did not truly use the same script the entire time. Like all writing systems the script did change as people simplified characters for their own convenience and as spoken (and thus written) language changed. New characters were added over time and old characters fell out of use. Cuneiform originally developed from commonly used pictograms that were used for various purposes (accounting, identification, etc.) but were not "true" writing. Same thing happened in China and Mesoamerica. The degree to which people were aware of the change in the characters is debatable, but there was certainly some awareness among the scribes and elites in later centuries (e.g. the Talmud records a debate about the Hebrew writing system). I suspect that at least the elites of Mesopotamia knew that the system had changed because they had access to much older tablets than what they themselves were producing. At a minimum they knew that the language had changed, since they were literate in both ancient Sumerian and whatever the common language of their city was (Akkadian, Elamite, etc.), and they likely recognized regional variations in the characters on tablets they received from other cities.