Note: There are already several related questions, without any definite answer.

I want to find an example of a Noetherian integral scheme $X$ which contains a Cartier divisor that is not a difference of two effective Cartier divisors. It will be great if $X$ is a variety.

This stack project exercise suggests a proper variety that is not projective. However, I cannot prove why this is a counterexample. It is not hard to show that $X$ does not have any nontrivial effective Cartier divisor. Hence it is enough to find an example of a nontrivial Cartier divisor, or a nontrivial line bundle, on $X$, but it is hard to construct such an example.