If it looks dirty and scummy, toss it. As Gonzo said, cut it in thirds, if I'm just spot claying, I'll cut it in fourths or even smaller. I tosss each piece after each use.
I use
clay magic and just store it in a ziplock bag. I don't spray any lube in there, depending on the clay and which lube, it could break down the clay. Just keep the air away from it and it should be fine.
Michael