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Yes. Note that the normal bundle of $S$ in $M$ is an oriented 1-dimensional bundle, in a natural way (by "dividing" the orientation from $M$ by the orientation from $S$). This implies that its restriction to $\gamma$ is trivial, so you can choose a non-zero section $X$. Now you can define the isotopy $\gamma_s(t) = Exp_{\gamma(t)}(sX(t))$ for some choice of Riemannian metric say, and for $s$ in a small enough neighborhood of $0$ this gives you a disjoint isotopic curve.