While what you say is true for Sanskrit is true, that Sanskrit is in a way fixed, it doesn't mean Sanskrit stopped evolving, it has certainly evolved since the time of Panini, just evolved in a way that doesn't break the rules set up by Panini. That however is a topic for another discussion as the vedas aren't composed in Paninian Sanskrit.

As for the vedas, specifically Rigveda (which is the only one I can comment on with some expertise) we have what we can call different recensions of the text. These texts, independently transmitted, differ very little from each other. This fact alone strongly indicate that even while they've been orally transmitted for many generations, they are more or less uncorrupted by the act of transmission.

Much more could be written on this subject (and has been). However, english isn't my native language, it is late and this comment could go on and on…short story is: memorization and oral transmission seems to work!