You could try to re-glaze the sink. There are kits and companies who do this sort of thing. The other stuff you try will not stick, and become stained and dirty looking.
Sounds like the sink got damaged or was a bad cast. Is it totally ceramic or is there some steel or cast iron? Give it a rap and see if it is tinny sounding.
If it is steel try to give it a push down at the drain, if it breaks the coating you can always get it refinished.