Riemann-Roch for nodal curves

Question

Let $X$ be an irreducible, nodal curve and $E$ a coherent subsheaf of a free sheaf $\oplus_{i=1}^r \mathcal{O}_X$ on $X$ of rank strictly less than $r$. Assume that $r \ge 2$. It follows that $H^0(E)$ are constant functions. Can we conclude that $\deg(E) \le 0$? I think this is true if $E$ is locally free. Am I right?