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(edited) Indeed Lazarsfeld's example mentioned by user132885 answers your first question: take $X=Y\times Z$ and $\mathcal{O}_X(D)=L\boxtimes M$, with $L$ ample and $M$ torsion, say of order $p$. Then $h^0(mD)=h^0(L^m)$ if $m$ is a multiple of $p$, and $0$ otherwise. The Iitaka dimension of $D$ is $\dim Y$.