ag.algebraic geometry - Torsion line bundles with non-vanishing cohomology on smooth ACM surfaces

I am looking for an example of a smooth surface $X$ with a fixed very ample $mathcal O_X(1)$ such that $H^1(mathcal O(k))=0$ for all $k$
(such thing is called an ACM surface, I think) and a globally generated line bundle $L$ such that $L$ is torsion in $Pic(X)$ and $H^1(L) neq 0$.

Does such surface exist? How can I construct one if it does exist? What if one ask for even nicer surface, such as arithmetically Gorenstein? If not, then I am willing to drop smooth or globally generated, but would like to keep the torsion condition.

More motivations(thanks Andrew): Such a line bundle would give a cyclic cover of $X$ which is not ACM, which would be of interest to me. I suppose one can think of this as a special counter example to a weaker (CM) version of purity of branch locus.

To the best of my knowledge this is not a homework question (: But I do not know much geometry, so may be some one can tell me where to find an answer. Thanks.

EDIT: Removed the global generation condition, by Dmitri's answer. I realized I did not really need it that much.

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