Let $X$ be a Noetherian, integral, separated, CM (Cohen-Macaulay) scheme.
Is it true that the set $\{ [L] \in Pic (X) : L$ is CM $ \}$ is finite ? If this is not true in general, then what if we also assume $\mathcal O_{X,x}$ is UFD for every $x \in X$ ?
Terminology: A coherent sheaf $L$ on a Noetherian scheme $X$ is called CM (Cohen-Macaulay) iff depth$_{\mathcal O_{X,x} } L_x =\dim_{\mathcal O_{X,x} } L_x , \forall x \in X$. A Noetherian scheme $X$ is called CM if its structure sheaf $\mathcal O_X$ is Cohen-Macaulay.
On a regular scheme, every nonzero locally free sheaf of finite rank is CM. So in particular, all line bundles on a regular scheme are Cohen-Macaulay. To find a counterexample to your claim (and your hoped-for improvement), it suffices to find a regular scheme with infinite Picard group. This is easy enough: take $\Bbb P^1$, for instance, with $Pic(\Bbb P^1)\cong \Bbb Z$.