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Here is an example. Let $X=\mathbb CP^2\times \mathbb CP^1$ blown up in one point. Denote by $E$ the exceptional divisor, and denote by $\pi$ the projection of $X$ to $\mathbb CP^2$. Then for $n$ sufficiently large $L_n=\pi^*(O(n))\otimes O(E)$ is what you want. More precisely, for each ample bundle $A$ on $X$ there is $n(A)\in \mathbb N$ such that $L_{n(A)}$ satisfies your conditions (we use $E^3=1$). Note that $L_{n}$ is never big.